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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Cauchy-Schwarz Inequality</a></div></nav><h1 class="article-title">Cauchy-Schwarz Inequality</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:26:38.879Z">Mar 01, 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Cauchy-Schwarz Inequality</a></div></nav><h1 class="article-title">Cauchy-Schwarz Inequality</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.378Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
 <blockquote class="callout example" data-callout="example">
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 <!DOCTYPE html>
-<html lang="en"><head><title>Cauchy Sequence</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Cauchy Sequence"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Cauchy Sequence"/><meta name="twitter:description" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta property="og:description" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Cauchy-Sequence"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Cauchy-Sequence"/><link rel="icon" href="../static/icon.png"/><meta name="description" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta name="generator" content="Quartz"/><link href="../index.css" rel="stylesheet" type="text/css" spa-preserve/><link href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" rel="stylesheet" type="text/css" spa-preserve/><script src="../prescript.js" type="application/javascript" spa-preserve></script><script type="application/javascript" spa-preserve>const fetchData = fetch("../static/contentIndex.json").then(data => data.json())</script></head><body data-slug="Definitions/Cauchy-Sequence"><div id="quartz-root" class="page"><div id="quartz-body"><div class="left sidebar"><h2 class="page-title"><a href="..">🪴 Quartz 4.0</a></h2><div 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Cauchy Sequence</a></div></nav><h1 class="article-title">Cauchy Sequence</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" 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+<html lang="en"><head><title>Cauchy Sequence</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Cauchy Sequence"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Cauchy Sequence"/><meta name="twitter:description" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta property="og:description" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Cauchy-Sequence"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Cauchy-Sequence"/><link rel="icon" href="../static/icon.png"/><meta name="description" content="Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy."/><meta name="generator" content="Quartz"/><link href="../index.css" rel="stylesheet" type="text/css" spa-preserve/><link href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" rel="stylesheet" type="text/css" spa-preserve/><script src="../prescript.js" type="application/javascript" spa-preserve></script><script type="application/javascript" spa-preserve>const fetchData = fetch("../static/contentIndex.json").then(data => data.json())</script></head><body data-slug="Definitions/Cauchy-Sequence"><div id="quartz-root" class="page"><div id="quartz-body"><div class="left sidebar"><h2 class="page-title"><a href="..">ACIT4330 Lecture 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Cauchy Sequence</a></div></nav><h1 class="article-title">Cauchy Sequence</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.377Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <strong>Cauchy sequence</strong> is a sequence where the elements become arbitrarily close to each other as the sequence progresses.</p>
 <h1 id="examples">Examples<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#examples" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="cauchy-sequence">Cauchy Sequence<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#cauchy-sequence" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
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-<html lang="en"><head><title>Characteristic Function</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Characteristic Function"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Characteristic Function"/><meta name="twitter:description" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta property="og:description" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" 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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0403em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span>, so <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0403em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> defined as:</p>
+<html lang="en"><head><title>Characteristic Function</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Characteristic Function"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Characteristic Function"/><meta name="twitter:description" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta property="og:description" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Functions/Characteristic-Function"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Functions/Characteristic-Function"/><link rel="icon" href="../../static/icon.png"/><meta name="description" content="\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}, so \varkappa_{y} : X \to \{ 0, \, 1 \} defined as: 1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y \end{cases}$$ ..."/><meta name="generator" content="Quartz"/><link href="../../index.css" rel="stylesheet" type="text/css" spa-preserve/><link href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" rel="stylesheet" type="text/css" spa-preserve/><script src="../../prescript.js" type="application/javascript" spa-preserve></script><script type="application/javascript" spa-preserve>const fetchData = fetch("../../static/contentIndex.json").then(data => data.json())</script></head><body data-slug="Definitions/Functions/Characteristic-Function"><div 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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/Functions/">Functions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Characteristic Function</a></div></nav><h1 class="article-title">Characteristic Function</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0403em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span>, so <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0403em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> defined as:</p>
 <span class="katex-error" title="ParseError: KaTeX parse error: Expected 'EOF', got '&amp;' at position 4: 1, &amp;̲ \text{if}\ x \…" style="color:#cc0000;">1, &amp; \text{if}\ x \in Y\\ 0, &amp; \text{if}\ x \notin Y
 \end{cases}$$</span></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span><br/>
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span><br/>
 where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, basically <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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diff --git a/public/Definitions/Functions/Inverse-Function.html b/public/Definitions/Functions/Inverse-Function.html
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 <!DOCTYPE html>
-<html lang="en"><head><title>Inverse Function</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Inverse Function"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Inverse Function"/><meta name="twitter:description" content="f^{-1} : Y \to X such that ff^{-1} = I_{x} \; \land f^{-1}f = I_{y} \exists f^{-1} \iff f \; \text{bijective}."/><meta property="og:description" content="f^{-1} : Y \to X such that ff^{-1} = I_{x} \; \land f^{-1}f = I_{y} \exists f^{-1} \iff f \; \text{bijective}."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="f^{-1} : Y \to X such that ff^{-1} = I_{x} \; \land f^{-1}f = I_{y} \exists f^{-1} \iff f \; \text{bijective}."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Functions/Inverse-Function"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Functions/Inverse-Function"/><link rel="icon" 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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><br/>
+<html lang="en"><head><title>Inverse Function</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Inverse Function"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Inverse Function"/><meta name="twitter:description" content="f^{-1} : Y \to X such that ff^{-1} = I_{x} \; \land f^{-1}f = I_{y} \exists f^{-1} \iff f \; \text{bijective}."/><meta 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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><br/>
 such that<br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">∃</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">bijective</span></span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A function which measures distances between two points in a <a href="../../Definitions/Metric-Spaces/Metric-Space" class="internal alias" data-slug="Definitions/Metric-Spaces/Metric-Space">Metric Space</a>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
                 s-3.542,0.634-4.898,1.688l-7.669-7.669C16.785,10.424,17,9.74,17,9c0-2.206-1.794-4-4-4S9,6.794,9,9s1.794,4,4,4
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 <!DOCTYPE html>
-<html lang="en"><head><title>Power Set</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Power Set"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Power Set"/><meta name="twitter:description" content="Definition \wp(X) of X consists of all the subsets of X. Amount of Elements in a Power Set Lets say we have |X|: |X| = |\{ 1, \, \dots, \, n \}| The \wp(X) would have 2^{n} elements in the set."/><meta property="og:description" content="Definition \wp(X) of X consists of all the subsets of X. Amount of Elements in a Power Set Lets say we have |X|: |X| = |\{ 1, \, \dots, \, n \}| The \wp(X) would have 2^{n} elements in the set."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition \wp(X) of X consists of all the subsets of X. 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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/Functions/">Functions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Power Set</a></div></nav><h1 class="article-title">Power Set</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">℘</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> consists of all the subsets of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.</p>
 <h2 id="amount-of-elements-in-a-power-set">Amount of Elements in a Power Set<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#amount-of-elements-in-a-power-set" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p>Lets say we have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span></span></span></span>:<br/>
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class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Functions/Characteristic-Function" class="internal">Characteristic Function</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Functions/Direct-Product" class="internal">Direct Product</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Functions/Inverse-Function" class="internal">Inverse Function</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time></p><div class="desc"><h3><a 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Hilbert Spaces</a></div></nav><h1 class="article-title">Hilbert Spaces</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Hilbert Spaces</a></div></nav><h1 class="article-title">Hilbert Spaces</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" 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 <p>These are <a href="../Banach-Space" class="internal alias" data-slug="Banach-Space">Banach Spaces</a> with norms given by an <a href="../Definitions/Inner-Product" class="internal alias" data-slug="Definitions/Inner-Product">Inner Product</a>.</p>
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diff --git a/public/Definitions/Inner-Product.html b/public/Definitions/Inner-Product.html
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--- a/public/Definitions/Inner-Product.html
+++ b/public/Definitions/Inner-Product.html
@@ -1,5 +1,5 @@
 <!DOCTYPE html>
-<html lang="en"><head><title>Inner Product</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Inner Product"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Inner Product"/><meta name="twitter:description" content="Definition (\cdot | \cdot) : V \times V \to \mathbb{C} Info V \times V \ni (u, v) \mapsto (u | v) \in \mathbb{C} Such that (au + bv | w) = a(u | w) + b(v|w) and \overline{(u | v)} = (v | u) (w | au ..."/><meta property="og:description" content="Definition (\cdot | \cdot) : V \times V \to \mathbb{C} Info V \times V \ni (u, v) \mapsto (u | v) \in \mathbb{C} Such that (au + bv | w) = a(u | w) + b(v|w) and \overline{(u | v)} = (v | u) (w | au ..."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition (\cdot | \cdot) : V \times V \to \mathbb{C} Info V \times V \ni (u, v) \mapsto (u | v) \in \mathbb{C} Such that (au + bv | w) = a(u | w) + b(v|w) and \overline{(u | v)} = (v | u) (w | au ..."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Inner Product</a></div></nav><h1 class="article-title">Inner Product</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" 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10.57.31.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw">Drawing 2025-02-13 11.02.27.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw">Drawing 2025-02-13 11.26.08.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw">Drawing 2025-02-13 11.47.33.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw">Drawing 2025-02-13 12.01.57.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw">Drawing 2025-02-13 12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Least Upper Bound Property</a></div></nav><h1 class="article-title">Least Upper Bound Property</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.378Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∃</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbb">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">such that</span></span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></p>
 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord mathnormal">p</span><span class="mopen">(⟨</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">⟩)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">⟩</span></span></span></span><br/>
diff --git a/public/Definitions/Linear-Map.html b/public/Definitions/Linear-Map.html
index fe9d1c5e..a2504faa 100644
--- a/public/Definitions/Linear-Map.html
+++ b/public/Definitions/Linear-Map.html
@@ -1,5 +1,5 @@
 <!DOCTYPE html>
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+<html lang="en"><head><title>Linear Map</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Linear Map"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Linear Map"/><meta name="twitter:description" content="Definition L: V \to \mathbb{C}^n \; \text{bijective} L (ax + by) = a \times L(x) + b \times L(y), \; \forall x, \, y \in V, \, a; \, b \in \mathbb{C} 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Linear Map</a></div></nav><h1 class="article-title">Linear Map</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbb">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">bijective</span></span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∥</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">∣</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/">Measure Theory</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/Sigma-Algebra/">Sigma Algebra</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Borel Measurable</a></div></nav><h1 class="article-title">Borel Measurable</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.368Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> is <a href="../../../Definitions/Topological-Spaces/Continuous" class="internal alias" data-slug="Definitions/Topological-Spaces/Continuous">continuous</a>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is Borel measurable.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
                 s-3.542,0.634-4.898,1.688l-7.669-7.669C16.785,10.424,17,9.74,17,9c0-2.206-1.794-4-4-4S9,6.794,9,9s1.794,4,4,4
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diff --git a/public/Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets.html b/public/Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets.html
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/">Measure Theory</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/Sigma-Algebra/">Sigma Algebra</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Borel Sets</a></div></nav><h1 class="article-title">Borel Sets</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.369Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 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 <p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-algebra generated by the <a href="../../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a> in a <a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological space</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-algebra of <strong>Borel Sets</strong> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/">Measure Theory</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/Sigma-Algebra/">Sigma Algebra</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Measurable</a></div></nav><h1 class="article-title">Measurable</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.369Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 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 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> (<a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">Topological Space</a>) is <strong>measurable</strong> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord text"><span class="mord"> open </span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../../../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../../../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../../../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/">Measure Theory</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/Sigma-Algebra/">Sigma Algebra</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Measure</a></div></nav><h1 class="article-title">Measure</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.369Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <strong>measure</strong> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∞</span><span class="mclose">]</span></span></span></span> such that:</p>
 <ol>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mopen">(</span><span class="mord">∅</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li>
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/">Measure Theory</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Measure-Theory/Sigma-Algebra/">Sigma Algebra</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Sigma-Algebra</a></div></nav><h1 class="article-title">Sigma-Algebra</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.369Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 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 <p>A <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>-algebra in a set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is a collection of subsets, so called measurable sets, of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> such that (the requirements are):</p>
 <ol>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span></li>
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diff --git a/public/Definitions/Metric-Spaces/Ball.html b/public/Definitions/Metric-Spaces/Ball.html
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 <!DOCTYPE html>
-<html lang="en"><head><title>Ball</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Ball"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Ball"/><meta name="twitter:description" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta property="og:description" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Metric-Spaces/Ball"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Metric-Spaces/Ball"/><link rel="icon" href="../../static/icon.png"/><meta name="description" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta name="generator" content="Quartz"/><link href="../../index.css" rel="stylesheet" type="text/css" spa-preserve/><link href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" rel="stylesheet" type="text/css" spa-preserve/><script src="../../prescript.js" type="application/javascript" spa-preserve></script><script type="application/javascript" spa-preserve>const fetchData = fetch("../../static/contentIndex.json").then(data => data.json())</script></head><body data-slug="Definitions/Metric-Spaces/Ball"><div id="quartz-root" class="page"><div id="quartz-body"><div class="left sidebar"><h2 class="page-title"><a href="../..">🪴 Quartz 4.0</a></h2><div class="spacer mobile-only"></div><div class="search"><button class="search-button" 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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
+<html lang="en"><head><title>Ball</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Ball"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Ball"/><meta name="twitter:description" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta property="og:description" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Metric-Spaces/Ball"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Metric-Spaces/Ball"/><link rel="icon" href="../../static/icon.png"/><meta name="description" content="Definition Let d be a metric on a set X. The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}. 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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> be a <a href="../../Definitions/Functions/Metric" class="internal alias" data-slug="Definitions/Functions/Metric">metric</a> on a set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.</p>
 <p>The (open) <strong>ball</strong> with centre <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> and radius <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose">}</span></span></span></span>.</p>
 <p>A sequence <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> <strong>converges</strong> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> if it eventually belongs to any ball <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>; <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.548em;vertical-align:-1.798em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.75em;"><span style="top:-1.377em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:-0.0502em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.75em;"><span class="svg-align" style="top:-2.102em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
diff --git a/public/Definitions/Metric-Spaces/Interior-Point.html b/public/Definitions/Metric-Spaces/Interior-Point.html
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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/Metric-Spaces/">Metric Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Metric Space</a></div></nav><h1 class="article-title">Metric Space</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <a href="../../Definitions/Functions/Metric" class="internal" data-slug="Definitions/Functions/Metric">Metric</a> on a set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∞</span><span class="mclose">⟩</span></span></span></span> such that</p>
 <ol>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></li>
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class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Metric-Spaces/Interior-Point" class="internal">Interior Point</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Metric-Spaces/Metric-Space" class="internal">Metric Space</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Metric-Spaces/Ball" class="internal">Ball</a></h3></div><ul class="tags"></ul></div></li></ul></div></div></div><hr/><div class="page-footer"></div></div><div class="right sidebar"></div><footer class><p>Created with <a href="https://quartz.jzhao.xyz/">Quartz v4.4.0</a> 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href="../Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="../Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="../Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="../Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="../Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="../Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="../Definitions/Linear-Map" 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9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Nets</a></div></nav><h1 class="article-title">Nets</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0763em;vertical-align:-1.3263em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1774em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">NET</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-1.6737em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">VFOs</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Nets</a></div></nav><h1 class="article-title">Nets</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.377Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0763em;vertical-align:-1.3263em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1774em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">NET</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-1.6737em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">VFOs</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.352em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
  35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
  0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
 -331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214
diff --git a/public/Definitions/Norm.html b/public/Definitions/Norm.html
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9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 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12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Norm</a></div></nav><h1 class="article-title">Norm</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <strong>norm</strong> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> is a map <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∞</span><span class="mclose">⟩</span></span></span></span> such that</p>
 <ol>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal">u</span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">∥</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span></li>
diff --git a/public/Definitions/Number-Field.html b/public/Definitions/Number-Field.html
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 <!DOCTYPE html>
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style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><br/>
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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Number Field</a></div></nav><h1 class="article-title">Number Field</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.372Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ba</span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7184em;vertical-align:-0.024em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Period of a Fraction</a></div></nav><h1 class="article-title">Period of a Fraction</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.377Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>The period of a fraction is the digits that repeat themselves.</p>
 <p>The digits can be from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">9</span></span></span></span>, and the length of the period is determined by the denominator, n in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> .</p>
 <h1 id="examples">Examples<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#examples" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 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12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Rational Cauchy Sequences</a></div></nav><h1 class="article-title">Rational Cauchy Sequences</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.367Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>They are sequences <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbb">Q</span><span class="mclose">}</span></span></span></span> such that<br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathbb">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span><br/>
 Such that<br/>
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+<html lang="en"><head><title>Complex Numbers</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Complex Numbers"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Complex Numbers"/><meta name="twitter:description" content="z = a + ib z = r(\cos \theta + i\sin \theta) z = re^{i \theta} Operations Multiplying Vectors z_{1} \times z_{2} = (r_{1} \times 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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ib</span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span><br/>
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 <p><a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">Open Sets</a> with union <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>An <strong>open map</strong> takes one <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open set</a> and maps it to another <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open set</a>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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diff --git a/public/Definitions/Sets/Open-Sets.html b/public/Definitions/Sets/Open-Sets.html
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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>An open set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span> (means set with a metric) consists only of <a href="../../Definitions/Metric-Spaces/Interior-Point" class="internal alias" data-slug="Definitions/Metric-Spaces/Interior-Point">interior points</a>.</p>
 <p>Then a sequence converges to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> it eventually belongs to any <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open set</a> containing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.368Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Sets/Open-Cover" class="internal">Open Cover</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.368Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Sets/Open-Map" class="internal">Open Map</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.368Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Sets/Open-Sets" class="internal">Open Sets</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.367Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Sets/Complex-Numbers" class="internal">Complex 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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span></p>
 <h1 id="truth-table">Truth Table<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#truth-table" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 
 
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span></p>
 <h1 id="truth-table">Truth Table<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#truth-table" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 
 
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><br/>
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><br/>
 The symbol (<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span>) is called negation</p>
 <h1 id="truth-table">Truth Table<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#truth-table" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 
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 <h1 id="truth-table">Truth Table<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#truth-table" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 
 
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href="../Definitions/Sets/Open-Sets" data-for="Definitions/Sets/Open-Sets">Open Sets</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Statements"><button class="folder-button"><span class="folder-title">Statements</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Statements"><li><a href="../Definitions/Statements/And" data-for="Definitions/Statements/And">And</a></li><li><a href="../Definitions/Statements/Implies" data-for="Definitions/Statements/Implies">Implies</a></li><li><a href="../Definitions/Statements/Not" data-for="Definitions/Statements/Not">Not</a></li><li><a href="../Definitions/Statements/Or" 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href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Subcover</a></div></nav><h1 class="article-title">Subcover</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" 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 <p>Any subcollection with union <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
                 s-3.542,0.634-4.898,1.688l-7.669-7.669C16.785,10.424,17,9.74,17,9c0-2.206-1.794-4-4-4S9,6.794,9,9s1.794,4,4,4
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12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Subnet</a></div></nav><h1 class="article-title">Subnet</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.378Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <strong>subnet</strong> of a <a href="../Definitions/Nets" class="internal alias" data-slug="Definitions/Nets">net</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is a <a href="../Definitions/Nets" class="internal alias" data-slug="Definitions/Nets">net</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> and a map <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∘</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> and such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∀</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>.</p>
 <h1 id="example">Example<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#example" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <blockquote class="callout example" data-callout="example">
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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="for-complex-numbers">For Complex Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#for-complex-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="for-complex-numbers">For Complex Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#for-complex-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Any equation<br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8144em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9641em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span><br/>
 with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> has <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> roots, or solutions, counted with multiplicity.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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-<html lang="en"><head><title>Bijective</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Bijective"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Bijective"/><meta name="twitter:description" content="Definition It is both Injective and Surjective. Bijective Map An example could be X = \{ 0,\, L \} \simeq \{ \text{Person}, \, \text{House} \} Where 0 would map to \text{Person} and L would map to \text{House}."/><meta property="og:description" content="Definition It is both Injective and Surjective. Bijective Map An example could be X = \{ 0,\, L \} \simeq \{ \text{Person}, \, \text{House} \} Where 0 would map to \text{Person} and L would map to \text{House}."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition It is both Injective and Surjective. 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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>It is both <a href="../../Definitions/Terminology/Injective" class="internal" data-slug="Definitions/Terminology/Injective">Injective</a> and <a href="../../Definitions/Terminology/Surjective" class="internal" data-slug="Definitions/Terminology/Surjective">Surjective</a>.</p>
 <h1 id="bijective-map">Bijective Map<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#bijective-map" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>An example could be<br/>
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 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∥</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>.<br/>
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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/Terminology/">Terminology</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Countable</a></div></nav><h1 class="article-title">Countable</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.372Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="what-is-countable">What is countable?<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#what-is-countable" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="natural-numbers">Natural Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#natural-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p>Quite simple<br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">}</span></span></span></span><br/>
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span><br/>
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span><br/>
 Usually a function would be mapped like:<br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">))</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">}</span></span></span></span></p>
 <h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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read</span></p></div></div><article class="popover-hint"><p>Can be written as “Q.E.D.” or “QED”.</p>
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read</span></p></div></div><article class="popover-hint"><p>Can be written as “Q.E.D.” or “QED”.</p>
 <p>It is shortened in Latin from “quod erat demonstrandum” (that which was to be demonstrated).</p>
 <h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A notation which is often placed at the end of a mathematical proof to indicate its completion</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span><br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><br/>
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diff --git a/public/Definitions/Topological-Spaces/Continuous.html b/public/Definitions/Topological-Spaces/Continuous.html
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-<html lang="en"><head><title>Continuous</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Continuous"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Continuous"/><meta name="twitter:description" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. We say f is open if f(B) is open and \forall open B \subset X."/><meta property="og:description" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. We say f is open if f(B) is open and \forall open B \subset X."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. We say f is open if f(B) is open and \forall open B \subset X."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Topological-Spaces/Continuous"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Topological-Spaces/Continuous"/><link rel="icon" href="../../static/icon.png"/><meta name="description" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. 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We say f is open if f(B) is open and \forall open B \subset X."/><meta property="og:description" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. We say f is open if f(B) is open and \forall open B \subset X."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. We say f is open if f(B) is open and \forall open B \subset X."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Definitions/Topological-Spaces/Continuous"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Definitions/Topological-Spaces/Continuous"/><link rel="icon" href="../../static/icon.png"/><meta name="description" content="Definition We say f is continuous (at every x) if f^{-1}(A) \equiv \{ x \in X | f(x) \in A \} is open for every open A \subset Y. 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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Continuous</a></div></nav><h1 class="article-title">Continuous</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.374Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>We say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is <strong>continuous</strong> (at every <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>) if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mclose">}</span></span></span></span> is open for every open <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>.</p>
 <p>We say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open</a> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> is <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open</a> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∀</span></span></span></span> <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.</p>
 <p>If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is a <a href="../../Definitions/Terminology/Bijective" class="internal alias" data-slug="Definitions/Terminology/Bijective">bijection</a> that is both <strong>continuous</strong> and <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open</a>, it is a <a href="../../Homeomorphic" class="internal alias" data-slug="Homeomorphic">homeomorphism</a>, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> are <a href="../../Homeomorphic" class="internal alias" data-slug="Homeomorphic">homeomorphic</a>, written <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≃</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>; they are the ‘same’ as <a href="../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological spaces</a>.</p>
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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>If there are two points <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> in a <a href="../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological space</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> that can be separated by neighbourhoods if there exists a neighbourhood <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> and a neighbourhood <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> are disjoint <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span>.</p>
 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is a <strong>Hausdorff space</strong> if any two distinct points in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> are separated by neighbourhoods.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Induced/">Induced</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Initial Topology</a></div></nav><h1 class="article-title">Initial Topology</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.375Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 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 <p>The <strong>initial topology</strong> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> induced by a family of functions <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> into <a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological spaces</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is the <a href="../../../Definitions/Topological-Spaces/Induced/Weakest-Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Induced/Weakest-Topology">weakest topology</a>on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> making all these functions <a href="../../../Definitions/Topological-Spaces/Continuous" class="internal alias" data-slug="Definitions/Topological-Spaces/Continuous">continuous</a>.</p>
 <p>Here: <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">open in</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Induced/">Induced</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Product Topology</a></div></nav><h1 class="article-title">Product Topology</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 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 <p>The <strong>product topology</strong> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord">Π</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological spaces</a>, is the <a href="../../../Definitions/Topological-Spaces/Induced/Initial-Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Induced/Initial-Topology">initial topology</a> induced by the family of projections <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Induced/">Induced</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Separating Points</a></div></nav><h1 class="article-title">Separating Points</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.375Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 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 <p>A family <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> of functions on a set <strong>separating points</strong> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> in the set if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> for some <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Induced/">Induced</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Weakest Topology</a></div></nav><h1 class="article-title">Weakest Topology</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.375Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 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 <p>Given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">℘</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>. The <strong>weakest topology</strong> on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> that contains <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> is the intersection of all the <a href="../../../Definitions/Topological-Spaces/Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Topology">topologies</a> that contains <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>. This is a <a href="../../../Definitions/Topological-Spaces/Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Topology">topology</a>, and consists of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>, and all unions of finite intersections of members from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>.</p>
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Terminologies/">Terminologies</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Compact</a></div></nav><h1 class="article-title">Compact</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 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diff --git a/public/Definitions/Topological-Spaces/Terminologies/Connected-Component.html b/public/Definitions/Topological-Spaces/Terminologies/Connected-Component.html
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href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Terminologies/">Terminologies</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Connected Component</a></div></nav><h1 class="article-title">Connected Component</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 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 <p>A <strong>connected component</strong> of a <a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological space</a> is the union of all <a href="../../../Definitions/Topological-Spaces/Terminologies/Connected" class="internal alias" data-slug="Definitions/Topological-Spaces/Terminologies/Connected">connected</a> subsets that contain a given point. It itself is <a href="../../../Definitions/Topological-Spaces/Terminologies/Connected" class="internal alias" data-slug="Definitions/Topological-Spaces/Terminologies/Connected">connected</a>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../../../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../../../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../../../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Terminologies/">Terminologies</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Connected</a></div></nav><h1 class="article-title">Connected</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological space</a> is <strong>connected</strong> if it is not a union of two non-empty <a href="../../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a>.</p>
 <p>i.e. if you draw the two non-empty <a href="../../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a> on the graph, if you have to lift your pen, it will not be connected.</p>
 <h1 id="examples">Examples<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#examples" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> be a topological space</p>
 <ol>
 <li>A <strong>neighbourhood</strong> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is an <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open set</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>.</li>
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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Topology</a></div></nav><h1 class="article-title">Topology</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.375Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A collection of subsets of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>, called <a href="../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a>, such that:</p>
 <ol>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∅</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span></li>
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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p>Can be abbreviated as “THM”.</p>
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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><p>Can be abbreviated as “THM”.</p>
 <h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>The product of compact spaces is compact in the <a href="../../Definitions/Topological-Spaces/Induced/Product-Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Induced/Product-Topology">product topology</a></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../../Definitions/Topological-Spaces/Terminologies" class="internal">Terminologies</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.876Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../../Definitions/Topological-Spaces/Induced" class="internal">Induced</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.875Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../../Definitions/Topological-Spaces/Hausdorff" class="internal">Hausdorff</a></h3></div><ul class="tags"></ul></div></li><li 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class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Topological-Spaces/Induced" class="internal">Induced</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Topological-Spaces/Terminologies" class="internal">Terminologies</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.375Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Topological-Spaces/Hausdorff" class="internal">Hausdorff</a></h3></div><ul class="tags"></ul></div></li><li 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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <strong>complex vector space</strong> is a set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> with addition <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> of vectors <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span>, and scalar multiplication <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;vertical-align:-0.1944em;"></span><span class="mord mathbb">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>, satisfying the <a href="../../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" class="internal alias" data-slug="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <strong>linear basis</strong> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> is a subset <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> such that every vector can be written uniquely as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for finitely many non-zero <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Have a metric given by the <a href="../../Definitions/Norm" class="internal" data-slug="Definitions/Norm">Norm</a> on a vector space <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∞</span><span class="mclose">⟩</span></span></span></span>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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diff --git a/public/Definitions/Vector-Spaces/Properties-of-a-Vector-Space.html b/public/Definitions/Vector-Spaces/Properties-of-a-Vector-Space.html
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+++ b/public/Definitions/Vector-Spaces/Properties-of-a-Vector-Space.html
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datetime="2025-03-01T13:37:40.374Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><ul>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</li>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mclose">)</span></span></span></span>,</li>
 <li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>,</li>
diff --git a/public/Definitions/Vector-Spaces/index.html b/public/Definitions/Vector-Spaces/index.html
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--- a/public/Definitions/Vector-Spaces/index.html
+++ b/public/Definitions/Vector-Spaces/index.html
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class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.874Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../../Definitions/Vector-Spaces/Complex-Vector-Space" class="internal">Complex Vector Space</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.874Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../../Definitions/Vector-Spaces/Linear-Basis" class="internal">Linear Basis</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.874Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../../Definitions/Vector-Spaces/Normed-Vector-Space" class="internal">Normed Vector Space</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.874Z">Mar 01, 2025</time></p><div 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class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.374Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Vector-Spaces/Normed-Vector-Space" class="internal">Normed Vector Space</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.374Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" class="internal">Properties of a Vector Space</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.373Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../../Definitions/Vector-Spaces/Complex-Vector-Space" class="internal">Complex Vector Space</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.373Z">01 Mar 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Definitions/">Definitions</a></div></nav><h1 class="article-title">Folder: Definitions</h1></div></div><div class="popover-hint"><article class></article><div class="page-listing"><p>21 items under this folder.</p><div><ul class="section-ul"><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.879Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Cauchy-Schwarz-Inequality" class="internal">Cauchy-Schwarz Inequality</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.879Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Least-Upper-Bound-Property" class="internal">Least Upper Bound Property</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.878Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Subnet" class="internal">Subnet</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.878Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Statements" class="internal">Statements</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Cauchy-Sequence" class="internal">Cauchy Sequence</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Nets" class="internal">Nets</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Period-of-a-Fraction" class="internal">Period of a Fraction</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.877Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Topological-Spaces" class="internal">Topological-Spaces</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.874Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Vector-Spaces" class="internal">Vector-Spaces</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.873Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Terminology" class="internal">Terminology</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Hilbert-Spaces" class="internal">Hilbert Spaces</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Inner-Product" class="internal">Inner Product</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Linear-Map" class="internal">Linear Map</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Norm" class="internal">Norm</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Number-Field" class="internal">Number Field</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Subcover" class="internal">Subcover</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.871Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Metric-Spaces" class="internal">Metric-Spaces</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.869Z">Mar 01, 2025</time></p><div class="desc"><h3><a href="../Definitions/Functions" class="internal">Functions</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:26:38.868Z">Mar 01, 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href="../Definitions/Cauchy-Sequence" class="internal">Cauchy Sequence</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.377Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Definitions/Nets" class="internal">Nets</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.377Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Definitions/Period-of-a-Fraction" class="internal">Period of a Fraction</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.376Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Definitions/Topological-Spaces" class="internal">Topological-Spaces</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li 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class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Definitions/Subcover" class="internal">Subcover</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.371Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Definitions/Metric-Spaces" class="internal">Metric-Spaces</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.370Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Definitions/Functions" class="internal">Functions</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/folder">folder</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.369Z">01 Mar 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diff --git a/public/Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw.html b/public/Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw.html
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datetime="2025-03-01T13:27:35.854Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. 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datetime="2025-03-01T13:37:40.385Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="G5MsgMXW">x </p>
diff --git a/public/Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw.html b/public/Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw.html
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datetime="2025-03-01T13:27:35.851Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
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datetime="2025-03-01T13:37:40.382Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="BbDzBXD0">x </p>
diff --git a/public/Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw.html b/public/Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw.html
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href="../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Excalidraw/">Excalidraw</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../Excalidraw/Lecture-11/">Lecture 11</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Drawing 2025-02-13 11.26.08.excalidraw</a></div></nav><h1 class="article-title">Drawing 2025-02-13 11.26.08.excalidraw</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.386Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="YhYKVuA6">x </p>
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datetime="2025-03-01T13:27:35.852Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
+<html lang="en"><head><title>Drawing 2025-02-13 11.47.33.excalidraw</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Drawing 2025-02-13 11.47.33.excalidraw"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Drawing 2025-02-13 11.47.33.excalidraw"/><meta name="twitter:description" content="⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this 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datetime="2025-03-01T13:37:40.382Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="aR0Ux44e">1 </p>
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datetime="2025-03-01T13:27:35.854Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
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                 s-3.542,0.634-4.898,1.688l-7.669-7.669C16.785,10.424,17,9.74,17,9c0-2.206-1.794-4-4-4S9,6.794,9,9s1.794,4,4,4
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datetime="2025-03-01T13:27:35.852Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
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datetime="2025-03-01T13:37:40.383Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. 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 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="9UNU6kFZ">0 </p>
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datetime="2025-03-01T13:37:40.382Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="WRpD5RRk">p </p>
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datetime="2025-03-01T13:27:35.853Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
+<html lang="en"><head><title>Drawing 2025-02-13 14.06.25.excalidraw</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Drawing 2025-02-13 14.06.25.excalidraw"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Drawing 2025-02-13 14.06.25.excalidraw"/><meta name="twitter:description" content="⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this 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datetime="2025-03-01T13:37:40.384Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. 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datetime="2025-03-01T13:37:40.380Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:1,&quot;scale&quot;:1.1,&quot;repelForce&quot;:0.5,&quot;centerForce&quot;:0.3,&quot;linkDistance&quot;:30,&quot;fontSize&quot;:0.6,&quot;opacityScale&quot;:1,&quot;showTags&quot;:true,&quot;removeTags&quot;:[],&quot;focusOnHover&quot;:false,&quot;enableRadial&quot;:false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
                 s-3.542,0.634-4.898,1.688l-7.669-7.669C16.785,10.424,17,9.74,17,9c0-2.206-1.794-4-4-4S9,6.794,9,9s1.794,4,4,4
diff --git a/public/Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw.html b/public/Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw.html
index f8171d0d..39d4a5ed 100644
--- a/public/Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw.html
+++ b/public/Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw.html
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datetime="2025-03-01T13:27:35.848Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
+<html lang="en"><head><title>Drawing 2025-02-24 12.47.32.excalidraw</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Drawing 2025-02-24 12.47.32.excalidraw"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Drawing 2025-02-24 12.47.32.excalidraw"/><meta name="twitter:description" content="⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this 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datetime="2025-03-01T13:27:35.850Z">Mar 01, 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. 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datetime="2025-03-01T13:37:40.386Z">01 Mar 2025</time><span>1 min read</span></p><ul class="tags"><li><a href="../../tags/excalidraw" class="internal tag-link">excalidraw</a></li></ul></div></div><article class="popover-hint"><p><span class="text-highlight">⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠</span> You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’</p>
 <h1 id="excalidraw-data">Excalidraw Data<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#excalidraw-data" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="text-elements">Text Elements<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#text-elements" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p id="8WbDf4Fn">A </p>
diff --git a/public/Excalidraw/Lecture-13/index.html b/public/Excalidraw/Lecture-13/index.html
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--- a/public/Excalidraw/Lecture-13/index.html
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href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a 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-<html lang="en"><head><title>Lecture 1 - 1.1 Sets and Numbers</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Lecture 1 - 1.1 Sets and Numbers"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 1 - 1.1 Sets and Numbers"/><meta name="twitter:description" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta property="og:description" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-1---1.1-Sets-and-Numbers"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-1---1.1-Sets-and-Numbers"/><link rel="icon" href="../static/icon.png"/><meta name="description" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta name="generator" content="Quartz"/><link href="../index.css" rel="stylesheet" type="text/css" spa-preserve/><link href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" rel="stylesheet" type="text/css" spa-preserve/><script src="../prescript.js" type="application/javascript" spa-preserve></script><script type="application/javascript" spa-preserve>const fetchData = fetch("../static/contentIndex.json").then(data => data.json())</script></head><body 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 1 - 1.1 Sets and Numbers</a></div></nav><h1 class="article-title">Lecture 1 - 1.1 Sets and Numbers</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-06T00:00:00.000Z">Jan 06, 2025</time><span>4 min read</span></p></div></div><article class="popover-hint"><h1 id="prime-numbers">Prime Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#prime-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" 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+<html lang="en"><head><title>Lecture 1 - 1.1 Sets and Numbers</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Lecture 1 - 1.1 Sets and Numbers"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 1 - 1.1 Sets and Numbers"/><meta name="twitter:description" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta property="og:description" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-1---1.1-Sets-and-Numbers"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-1---1.1-Sets-and-Numbers"/><link rel="icon" href="../static/icon.png"/><meta name="description" content="Prime Numbers \mathbb{N} = \{1,2,3,\dots\} (natural numbers) Prime Number - p \in \mathbb{N} \setminus \{1\} only divisible by 1 and p. They are building blocks for multiplication; for instance 90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2 p \times q = p' \times q' \implies (need proof Theorem 1.1.1) p = p' \; \cap \; q=q' p = q' \; \cap \; q = p' Theorem 1.1.1 Any natural number other than one is a product of unique primes Proof Existence Divide as long as possible Uniqueness (Gauss): Need Euclid’s lemma, saying that If n|ab with gcd(a,b) = 1, then n|a or n|b."/><meta name="generator" content="Quartz"/><link href="../index.css" rel="stylesheet" type="text/css" spa-preserve/><link href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" rel="stylesheet" type="text/css" spa-preserve/><script src="../prescript.js" type="application/javascript" spa-preserve></script><script type="application/javascript" spa-preserve>const fetchData = fetch("../static/contentIndex.json").then(data => data.json())</script></head><body 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 1 - 1.1 Sets and Numbers</a></div></nav><h1 class="article-title">Lecture 1 - 1.1 Sets and Numbers</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-06T00:00:00.000Z">06 Jan 2025</time><span>4 min read</span></p></div></div><article class="popover-hint"><h1 id="prime-numbers">Prime Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#prime-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">}</span></span></span></span> (natural numbers)</p>
 <p><strong>Prime Number</strong> - <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbb">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mclose">}</span></span></span></span> only divisible by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span>.</p>
 <p>They are building blocks for multiplication; for instance<br/>
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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 11</a></div></nav><h1 class="article-title">Lecture 11</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-13T00:00:00.000Z">Feb 13, 2025</time><span>5 min read</span></p></div></div><article class="popover-hint"><p>Last lecture talked about <a href="../Definitions/Nets" class="internal" data-slug="Definitions/Nets">Nets</a></p>
+<html lang="en"><head><title>Lecture 11</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Lecture 11"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 11"/><meta name="twitter:description" content="Last lecture talked about Nets Proposition A topological space is Hausdorff \iff each net converges to at most one point. Proof \Rightarrow: “Easy” \Leftarrow: say x \neq y, therefore cannot be separated by disjoint neighbourhoods."/><meta property="og:description" content="Last lecture talked about Nets Proposition A topological space is Hausdorff \iff each net converges to at most one point. Proof \Rightarrow: “Easy” \Leftarrow: say x \neq y, therefore cannot be separated by disjoint neighbourhoods."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Last lecture talked about Nets Proposition A topological space is Hausdorff \iff each net converges to at most one point. Proof \Rightarrow: “Easy” \Leftarrow: say x \neq y, therefore cannot be separated by disjoint neighbourhoods."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta name="twitter:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="og:image" content="https://quartz.jzhao.xyz/static/og-image.png"/><meta property="twitter:domain" content="quartz.jzhao.xyz"/><meta property="og:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-11"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-11"/><link rel="icon" href="../static/icon.png"/><meta name="description" content="Last lecture talked about Nets Proposition A topological space is Hausdorff \iff each net converges to at most one point. 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 11</a></div></nav><h1 class="article-title">Lecture 11</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-13T00:00:00.000Z">13 Feb 2025</time><span>5 min read</span></p></div></div><article class="popover-hint"><p>Last lecture talked about <a href="../Definitions/Nets" class="internal" data-slug="Definitions/Nets">Nets</a></p>
 <h1 id="proposition">Proposition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#proposition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>A <a href="../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological space</a> is <a href="../Definitions/Topological-Spaces/Hausdorff" class="internal" data-slug="Definitions/Topological-Spaces/Hausdorff">Hausdorff</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> each <a href="../Definitions/Nets" class="internal alias" data-slug="Definitions/Nets">net</a> converges to at most one point.</p>
 <h2 id="proof">Proof<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#proof" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 12 - Induced Topologies</a></div></nav><h1 class="article-title">Lecture 12 - Induced Topologies</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-24T00:00:00.000Z">24 Feb 2025</time><span>3 min read</span></p></div></div><article class="popover-hint"><h1 id="weakest-topology">Weakest Topology<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#weakest-topology" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Definition of <a href="../Definitions/Topological-Spaces/Induced/Weakest-Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Induced/Weakest-Topology">Weakest Topology</a> defined in the lecture.</p>
 <h1 id="initial-topology">Initial Topology<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#initial-topology" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Definition of <a href="../Definitions/Topological-Spaces/Induced/Initial-Topology" class="internal alias" data-slug="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a>, defined in the lecture.</p>
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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 13 - Measure Theory</a></div></nav><h1 class="article-title">Lecture 13 - Measure Theory</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-27T00:00:00.000Z">Feb 27, 2025</time><span>5 min read</span></p></div></div><article class="popover-hint"><h1 id="sigma-algebra">Sigma Algebra<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#sigma-algebra" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 13 - Measure Theory</a></div></nav><h1 class="article-title">Lecture 13 - Measure Theory</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-27T00:00:00.000Z">27 Feb 2025</time><span>5 min read</span></p></div></div><article class="popover-hint"><h1 id="sigma-algebra">Sigma Algebra<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#sigma-algebra" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Defined in the lecture: <a href="../Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra" class="internal" data-slug="Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra">Sigma-Algebra</a><br/>
 (See also <a href="../Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra#measurable" class="internal alias" data-slug="Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra">Measurable</a>)</p>
 <h2 id="measure">Measure<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#measure" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
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href="../Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="../Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="../Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="../Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="../Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="../Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="../Definitions/Linear-Map" 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 2</a></div></nav><h1 class="article-title">Lecture 2</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-09T00:00:00.000Z">09 Jan 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><h1 id="quotients">Quotients<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#quotients" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="constructing-quotients">Constructing Quotients<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#constructing-quotients" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p>Have equivalence relation on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7722em;vertical-align:-0.0833em;"></span><span class="mord mathbb">Z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathbb">Z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">})</span></span></span></span> given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span> when <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">c</span></span></span></span>.</p>
 <p>Then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.355em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">∼</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbb mtight">Z</span><span class="mbin mtight">×</span><span class="mord mathbb mtight">Z</span><span class="mbin mtight">∖</span><span class="mopen mtight">{</span><span class="mord mtight">0</span><span class="mclose mtight">}</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> is the set of rational numbers <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8556em;vertical-align:-0.1667em;"></span><span class="mord mathbb">Q</span></span></span></span> with addition and multiplication defined in an <strong>obvious</strong> way.<br/>
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-<html lang="en"><head><title>Lecture 3</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="🪴 Quartz 4.0"/><meta property="og:title" content="Lecture 3"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 3"/><meta name="twitter:description" content="Proposition 1.1.4 Each real number is a limit of a sequence of rational numbers. \mathbb{Q} \subset \mathbb{R} As an ordered Number Field \mathbb{R} is complete, 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 3</a></div></nav><h1 class="article-title">Lecture 3</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-13T00:00:00.000Z">Jan 13, 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><h1 id="proposition-114">Proposition 1.1.4<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#proposition-114" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 3</a></div></nav><h1 class="article-title">Lecture 3</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-13T00:00:00.000Z">13 Jan 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><h1 id="proposition-114">Proposition 1.1.4<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#proposition-114" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>Each real number is a limit of a sequence of rational numbers.<br/>
 <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8556em;vertical-align:-0.1667em;"></span><span class="mord mathbb">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span><br/>
 As an ordered <a href="../Definitions/Number-Field" class="internal alias" data-slug="Definitions/Number-Field">Number Field</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> is <strong>complete</strong>, meaning that every <a href="../Definitions/Cauchy-Sequence" class="internal alias" data-slug="Definitions/Cauchy-Sequence">Cauchy Sequence</a> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> converges to a real number.<br/>
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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 4 - 1.2 Metric Spaces</a></div></nav><h1 class="article-title">Lecture 4 - 1.2 Metric Spaces</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-16T00:00:00.000Z">16 Jan 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><h1 id="the-inverse-image">The Inverse Image<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#the-inverse-image" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>The Inverse Image uses a <a href="../Definitions/Functions/Inverse-Function" class="internal alias" data-slug="Definitions/Functions/Inverse-Function">Inverse Function</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> written <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">}</span></span></span></span>.</p>
 <h1 id="complex-numbers">Complex Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#complex-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>In <a href="../Definitions/Sets/Complex-Numbers" class="internal alias" data-slug="Definitions/Sets/Complex-Numbers">Complex Numbers</a>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7722em;vertical-align:-0.0833em;"></span><span class="mord mathbb">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> with usual addition of vectors</p>
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class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Measure-Theory/Sigma-Algebra"><button class="folder-button"><span class="folder-title">Sigma-Algebra</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Measure-Theory/Sigma-Algebra"><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable" data-for="Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable">Borel Measurable</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets" data-for="Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets">Borel Sets</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Measurable" data-for="Definitions/Measure-Theory/Sigma-Algebra/Measurable">Measurable</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Measure" data-for="Definitions/Measure-Theory/Sigma-Algebra/Measure">Measure</a></li><li><a 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data-folderul="Definitions/Topological-Spaces"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Induced"><button class="folder-button"><span class="folder-title">Induced</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Induced"><li><a href="../Definitions/Topological-Spaces/Induced/Initial-Topology" data-for="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Product-Topology" data-for="Definitions/Topological-Spaces/Induced/Product-Topology">Product Topology</a></li><li><a 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href="../Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="../Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="../Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="../Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="../Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="../Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="../Definitions/Linear-Map" 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12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 5</a></div></nav><h1 class="article-title">Lecture 5</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-20T00:00:00.000Z">Jan 20, 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><blockquote class="callout info" data-callout="info">
+<html lang="en"><head><title>Lecture 5</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Lecture 5"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 5"/><meta name="twitter:description" content="Info A normed space is a Banach Space when the corresponding metric space is complete. Any normed space can be completed to a Banach Space, see the real numbers from \mathbb{Q}."/><meta property="og:description" content="Info A normed space is a Banach Space when the corresponding metric space is complete. Any normed space can be completed to a Banach Space, see the real numbers from \mathbb{Q}."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Info A normed space is a Banach Space when the corresponding metric space is complete. 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9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 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12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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data-folderul="Definitions/Topological-Spaces"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Induced"><button class="folder-button"><span class="folder-title">Induced</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Induced"><li><a href="../Definitions/Topological-Spaces/Induced/Initial-Topology" data-for="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Product-Topology" data-for="Definitions/Topological-Spaces/Induced/Product-Topology">Product Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Separating-Points" data-for="Definitions/Topological-Spaces/Induced/Separating-Points">Separating Points</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Weakest-Topology" data-for="Definitions/Topological-Spaces/Induced/Weakest-Topology">Weakest Topology</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Terminologies"><button class="folder-button"><span class="folder-title">Terminologies</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Terminologies"><li><a href="../Definitions/Topological-Spaces/Terminologies/Compact" data-for="Definitions/Topological-Spaces/Terminologies/Compact">Compact</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected" data-for="Definitions/Topological-Spaces/Terminologies/Connected">Connected</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected-Component" data-for="Definitions/Topological-Spaces/Terminologies/Connected-Component">Connected Component</a></li></ul></div></li><li><a href="../Definitions/Topological-Spaces/Continuous" data-for="Definitions/Topological-Spaces/Continuous">Continuous</a></li><li><a href="../Definitions/Topological-Spaces/Hausdorff" data-for="Definitions/Topological-Spaces/Hausdorff">Hausdorff</a></li><li><a href="../Definitions/Topological-Spaces/Topological-Space" data-for="Definitions/Topological-Spaces/Topological-Space">Topological Space</a></li><li><a href="../Definitions/Topological-Spaces/Topology" data-for="Definitions/Topological-Spaces/Topology">Topology</a></li><li><a href="../Definitions/Topological-Spaces/Tychonoff-Theorem" data-for="Definitions/Topological-Spaces/Tychonoff-Theorem">Tychonoff Theorem</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Vector-Spaces"><button class="folder-button"><span class="folder-title">Vector Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Vector-Spaces"><li><a href="../Definitions/Vector-Spaces/Complex-Vector-Space" data-for="Definitions/Vector-Spaces/Complex-Vector-Space">Complex Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Linear-Basis" data-for="Definitions/Vector-Spaces/Linear-Basis">Linear Basis</a></li><li><a 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href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 6 - 2.1 Topology</a></div></nav><h1 class="article-title">Lecture 6 - 2.1 Topology</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-27T00:00:00.000Z">27 Jan 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="open-sets">Open Sets<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#open-sets" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <p>The concept of <a href="../Definitions/Metric-Spaces/Ball" class="internal alias" data-slug="Definitions/Metric-Spaces/Ball">balls</a> is required to understand what an <a href="../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a> are.</p>
 <h2 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <p>The definition can be found in <a href="../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">Open Sets</a>.<br/>
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class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span> ball topology<br/>
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class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Measure-Theory/Sigma-Algebra"><button class="folder-button"><span class="folder-title">Sigma-Algebra</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Measure-Theory/Sigma-Algebra"><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable" data-for="Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable">Borel Measurable</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets" data-for="Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets">Borel Sets</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Measurable" data-for="Definitions/Measure-Theory/Sigma-Algebra/Measurable">Measurable</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Measure" data-for="Definitions/Measure-Theory/Sigma-Algebra/Measure">Measure</a></li><li><a href="../Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra" data-for="Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra">Sigma-Algebra</a></li></ul></div></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Metric-Spaces"><button class="folder-button"><span class="folder-title">Metric Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Metric-Spaces"><li><a href="../Definitions/Metric-Spaces/Ball" data-for="Definitions/Metric-Spaces/Ball">Ball</a></li><li><a href="../Definitions/Metric-Spaces/Interior-Point" data-for="Definitions/Metric-Spaces/Interior-Point">Interior Point</a></li><li><a href="../Definitions/Metric-Spaces/Metric-Space" data-for="Definitions/Metric-Spaces/Metric-Space">Metric Space</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Sets"><button class="folder-button"><span class="folder-title">Sets</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Sets"><li><a href="../Definitions/Sets/Complex-Numbers" data-for="Definitions/Sets/Complex-Numbers">Complex Numbers</a></li><li><a href="../Definitions/Sets/Open-Cover" data-for="Definitions/Sets/Open-Cover">Open Cover</a></li><li><a href="../Definitions/Sets/Open-Map" data-for="Definitions/Sets/Open-Map">Open Map</a></li><li><a href="../Definitions/Sets/Open-Sets" data-for="Definitions/Sets/Open-Sets">Open Sets</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Statements"><button class="folder-button"><span class="folder-title">Statements</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Statements"><li><a href="../Definitions/Statements/And" data-for="Definitions/Statements/And">And</a></li><li><a href="../Definitions/Statements/Implies" data-for="Definitions/Statements/Implies">Implies</a></li><li><a href="../Definitions/Statements/Not" data-for="Definitions/Statements/Not">Not</a></li><li><a href="../Definitions/Statements/Or" data-for="Definitions/Statements/Or">Or</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Terminology"><button class="folder-button"><span class="folder-title">Terminology</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Terminology"><li><a href="../Definitions/Terminology/Algebraically-Complete" data-for="Definitions/Terminology/Algebraically-Complete">Algebraically Complete</a></li><li><a href="../Definitions/Terminology/Bijective" data-for="Definitions/Terminology/Bijective">Bijective</a></li><li><a href="../Definitions/Terminology/Bounded" data-for="Definitions/Terminology/Bounded">Bounded</a></li><li><a href="../Definitions/Terminology/Countable" data-for="Definitions/Terminology/Countable">Countable</a></li><li><a href="../Definitions/Terminology/Injective" data-for="Definitions/Terminology/Injective">Injective</a></li><li><a href="../Definitions/Terminology/QED" data-for="Definitions/Terminology/QED">QED</a></li><li><a href="../Definitions/Terminology/Surjective" data-for="Definitions/Terminology/Surjective">Surjective</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces"><button class="folder-button"><span class="folder-title">Topological Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Induced"><button class="folder-button"><span class="folder-title">Induced</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Induced"><li><a href="../Definitions/Topological-Spaces/Induced/Initial-Topology" data-for="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Product-Topology" data-for="Definitions/Topological-Spaces/Induced/Product-Topology">Product Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Separating-Points" data-for="Definitions/Topological-Spaces/Induced/Separating-Points">Separating Points</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Weakest-Topology" data-for="Definitions/Topological-Spaces/Induced/Weakest-Topology">Weakest Topology</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Terminologies"><button class="folder-button"><span class="folder-title">Terminologies</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Terminologies"><li><a href="../Definitions/Topological-Spaces/Terminologies/Compact" data-for="Definitions/Topological-Spaces/Terminologies/Compact">Compact</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected" data-for="Definitions/Topological-Spaces/Terminologies/Connected">Connected</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected-Component" data-for="Definitions/Topological-Spaces/Terminologies/Connected-Component">Connected Component</a></li></ul></div></li><li><a href="../Definitions/Topological-Spaces/Continuous" data-for="Definitions/Topological-Spaces/Continuous">Continuous</a></li><li><a href="../Definitions/Topological-Spaces/Hausdorff" data-for="Definitions/Topological-Spaces/Hausdorff">Hausdorff</a></li><li><a href="../Definitions/Topological-Spaces/Topological-Space" data-for="Definitions/Topological-Spaces/Topological-Space">Topological Space</a></li><li><a href="../Definitions/Topological-Spaces/Topology" data-for="Definitions/Topological-Spaces/Topology">Topology</a></li><li><a href="../Definitions/Topological-Spaces/Tychonoff-Theorem" data-for="Definitions/Topological-Spaces/Tychonoff-Theorem">Tychonoff Theorem</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Vector-Spaces"><button class="folder-button"><span class="folder-title">Vector Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Vector-Spaces"><li><a href="../Definitions/Vector-Spaces/Complex-Vector-Space" data-for="Definitions/Vector-Spaces/Complex-Vector-Space">Complex Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Linear-Basis" data-for="Definitions/Vector-Spaces/Linear-Basis">Linear Basis</a></li><li><a href="../Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="../Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="../Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="../Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="../Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="../Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="../Definitions/Linear-Map" data-for="Definitions/Linear-Map">Linear Map</a></li><li><a href="../Definitions/Nets" data-for="Definitions/Nets">Nets</a></li><li><a href="../Definitions/Norm" data-for="Definitions/Norm">Norm</a></li><li><a href="../Definitions/Number-Field" data-for="Definitions/Number-Field">Number Field</a></li><li><a href="../Definitions/Period-of-a-Fraction" data-for="Definitions/Period-of-a-Fraction">Period of a Fraction</a></li><li><a href="../Definitions/Rational-Cauchy-Sequences" data-for="Definitions/Rational-Cauchy-Sequences">Rational Cauchy Sequences</a></li><li><a href="../Definitions/Subcover" data-for="Definitions/Subcover">Subcover</a></li><li><a href="../Definitions/Subnet" data-for="Definitions/Subnet">Subnet</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 10.57.31.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw">Drawing 2025-02-13 11.02.27.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw">Drawing 2025-02-13 11.26.08.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw">Drawing 2025-02-13 11.47.33.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw">Drawing 2025-02-13 12.01.57.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw">Drawing 2025-02-13 12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw" data-for="Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw">Drawing 2025-02-24 11.58.37.excalidraw</a></li><li><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw" data-for="Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw">Drawing 2025-02-24 12.47.32.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-13"><button class="folder-button"><span class="folder-title">Lecture 13</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-13"><li><a href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 7</a></div></nav><h1 class="article-title">Lecture 7</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-30T00:00:00.000Z">30 Jan 2025</time><span>4 min read</span></p></div></div><article class="popover-hint"><p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span> ball topology<br/>
 Say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> is a closed set in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> (in other words, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9405em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9405em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord amsrm mtight">∁</span></span></span></span></span></span></span></span></span></span></span></span> <a href="../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open</a>).</p>
 <p>Say we have a sequence <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> and that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.<br/>
 Then <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>.</p>
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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 8</a></div></nav><h1 class="article-title">Lecture 8</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-03T00:00:00.000Z">Feb 03, 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><p>Defines in <a href="../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological spaces</a>:</p>
+<html lang="en"><head><title>Lecture 8</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&amp;family=Schibsted Grotesk:wght@400;700&amp;family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&amp;display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Lecture 8"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 8"/><meta name="twitter:description" content="Defines in topological spaces: Connected Connected Component Proposition [0, 1] is connected. 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 8</a></div></nav><h1 class="article-title">Lecture 8</h1><p show-comma="true" class="content-meta"><time datetime="2025-02-03T00:00:00.000Z">03 Feb 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><p>Defines in <a href="../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological spaces</a>:</p>
 <ul>
 <li><a href="../Definitions/Topological-Spaces/Terminologies/Connected" class="internal" data-slug="Definitions/Topological-Spaces/Terminologies/Connected">Connected</a></li>
 <li><a href="../Definitions/Topological-Spaces/Terminologies/Connected-Component" class="internal alias" data-slug="Definitions/Topological-Spaces/Terminologies/Connected-Component">Connected Component</a></li>
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href="../Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra" data-for="Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra">Sigma-Algebra</a></li></ul></div></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Metric-Spaces"><button class="folder-button"><span class="folder-title">Metric Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Metric-Spaces"><li><a href="../Definitions/Metric-Spaces/Ball" data-for="Definitions/Metric-Spaces/Ball">Ball</a></li><li><a href="../Definitions/Metric-Spaces/Interior-Point" data-for="Definitions/Metric-Spaces/Interior-Point">Interior Point</a></li><li><a href="../Definitions/Metric-Spaces/Metric-Space" data-for="Definitions/Metric-Spaces/Metric-Space">Metric Space</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Sets"><button class="folder-button"><span class="folder-title">Sets</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Sets"><li><a href="../Definitions/Sets/Complex-Numbers" data-for="Definitions/Sets/Complex-Numbers">Complex Numbers</a></li><li><a href="../Definitions/Sets/Open-Cover" data-for="Definitions/Sets/Open-Cover">Open Cover</a></li><li><a href="../Definitions/Sets/Open-Map" data-for="Definitions/Sets/Open-Map">Open Map</a></li><li><a href="../Definitions/Sets/Open-Sets" data-for="Definitions/Sets/Open-Sets">Open Sets</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Statements"><button class="folder-button"><span class="folder-title">Statements</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Statements"><li><a href="../Definitions/Statements/And" data-for="Definitions/Statements/And">And</a></li><li><a href="../Definitions/Statements/Implies" data-for="Definitions/Statements/Implies">Implies</a></li><li><a href="../Definitions/Statements/Not" data-for="Definitions/Statements/Not">Not</a></li><li><a href="../Definitions/Statements/Or" data-for="Definitions/Statements/Or">Or</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Terminology"><button class="folder-button"><span class="folder-title">Terminology</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Terminology"><li><a href="../Definitions/Terminology/Algebraically-Complete" data-for="Definitions/Terminology/Algebraically-Complete">Algebraically Complete</a></li><li><a href="../Definitions/Terminology/Bijective" data-for="Definitions/Terminology/Bijective">Bijective</a></li><li><a href="../Definitions/Terminology/Bounded" data-for="Definitions/Terminology/Bounded">Bounded</a></li><li><a href="../Definitions/Terminology/Countable" data-for="Definitions/Terminology/Countable">Countable</a></li><li><a href="../Definitions/Terminology/Injective" data-for="Definitions/Terminology/Injective">Injective</a></li><li><a href="../Definitions/Terminology/QED" data-for="Definitions/Terminology/QED">QED</a></li><li><a href="../Definitions/Terminology/Surjective" data-for="Definitions/Terminology/Surjective">Surjective</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces"><button class="folder-button"><span class="folder-title">Topological Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Induced"><button class="folder-button"><span class="folder-title">Induced</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Induced"><li><a href="../Definitions/Topological-Spaces/Induced/Initial-Topology" data-for="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Product-Topology" data-for="Definitions/Topological-Spaces/Induced/Product-Topology">Product Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Separating-Points" data-for="Definitions/Topological-Spaces/Induced/Separating-Points">Separating Points</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Weakest-Topology" data-for="Definitions/Topological-Spaces/Induced/Weakest-Topology">Weakest Topology</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Terminologies"><button class="folder-button"><span class="folder-title">Terminologies</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Terminologies"><li><a href="../Definitions/Topological-Spaces/Terminologies/Compact" data-for="Definitions/Topological-Spaces/Terminologies/Compact">Compact</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected" data-for="Definitions/Topological-Spaces/Terminologies/Connected">Connected</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected-Component" data-for="Definitions/Topological-Spaces/Terminologies/Connected-Component">Connected Component</a></li></ul></div></li><li><a href="../Definitions/Topological-Spaces/Continuous" data-for="Definitions/Topological-Spaces/Continuous">Continuous</a></li><li><a href="../Definitions/Topological-Spaces/Hausdorff" data-for="Definitions/Topological-Spaces/Hausdorff">Hausdorff</a></li><li><a href="../Definitions/Topological-Spaces/Topological-Space" data-for="Definitions/Topological-Spaces/Topological-Space">Topological Space</a></li><li><a href="../Definitions/Topological-Spaces/Topology" data-for="Definitions/Topological-Spaces/Topology">Topology</a></li><li><a href="../Definitions/Topological-Spaces/Tychonoff-Theorem" data-for="Definitions/Topological-Spaces/Tychonoff-Theorem">Tychonoff Theorem</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Vector-Spaces"><button class="folder-button"><span class="folder-title">Vector Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Vector-Spaces"><li><a href="../Definitions/Vector-Spaces/Complex-Vector-Space" data-for="Definitions/Vector-Spaces/Complex-Vector-Space">Complex Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Linear-Basis" data-for="Definitions/Vector-Spaces/Linear-Basis">Linear Basis</a></li><li><a href="../Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="../Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="../Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="../Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="../Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="../Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="../Definitions/Linear-Map" data-for="Definitions/Linear-Map">Linear Map</a></li><li><a href="../Definitions/Nets" data-for="Definitions/Nets">Nets</a></li><li><a href="../Definitions/Norm" data-for="Definitions/Norm">Norm</a></li><li><a href="../Definitions/Number-Field" data-for="Definitions/Number-Field">Number Field</a></li><li><a href="../Definitions/Period-of-a-Fraction" data-for="Definitions/Period-of-a-Fraction">Period of a Fraction</a></li><li><a href="../Definitions/Rational-Cauchy-Sequences" data-for="Definitions/Rational-Cauchy-Sequences">Rational Cauchy Sequences</a></li><li><a href="../Definitions/Subcover" data-for="Definitions/Subcover">Subcover</a></li><li><a href="../Definitions/Subnet" data-for="Definitions/Subnet">Subnet</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 10.57.31.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw">Drawing 2025-02-13 11.02.27.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw">Drawing 2025-02-13 11.26.08.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw">Drawing 2025-02-13 11.47.33.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw">Drawing 2025-02-13 12.01.57.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw">Drawing 2025-02-13 12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw" data-for="Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw">Drawing 2025-02-24 11.58.37.excalidraw</a></li><li><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw" data-for="Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw">Drawing 2025-02-24 12.47.32.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-13"><button class="folder-button"><span class="folder-title">Lecture 13</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-13"><li><a href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a></div></nav><h1 class="article-title">Folder: Lectures</h1></div></div><div class="popover-hint"><article class></article><div class="page-listing"><p>11 items under this folder.</p><div><ul class="section-ul"><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-27T00:00:00.000Z">Feb 27, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-13---Measure-Theory" class="internal">Lecture 13 - Measure Theory</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-24T00:00:00.000Z">Feb 24, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-12---Induced-Topologies" class="internal">Lecture 12 - Induced Topologies</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-13T00:00:00.000Z">Feb 13, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-11" class="internal">Lecture 11</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-03T00:00:00.000Z">Feb 03, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-8" class="internal">Lecture 8</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-30T00:00:00.000Z">Jan 30, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-7" class="internal">Lecture 7</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-27T00:00:00.000Z">Jan 27, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-6---2.1-Topology" class="internal">Lecture 6 - 2.1 Topology</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-20T00:00:00.000Z">Jan 20, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-5" class="internal">Lecture 5</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-16T00:00:00.000Z">Jan 16, 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" class="internal">Lecture 4 - 1.2 Metric Spaces</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-13T00:00:00.000Z">Jan 13, 2025</time></p><div class="desc"><h3><a 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a></div></nav><h1 class="article-title">Folder: Lectures</h1></div></div><div class="popover-hint"><article class></article><div class="page-listing"><p>11 items under this folder.</p><div><ul class="section-ul"><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-27T00:00:00.000Z">27 Feb 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-13---Measure-Theory" class="internal">Lecture 13 - Measure Theory</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-24T00:00:00.000Z">24 Feb 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-12---Induced-Topologies" class="internal">Lecture 12 - Induced Topologies</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-13T00:00:00.000Z">13 Feb 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-11" class="internal">Lecture 11</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-02-03T00:00:00.000Z">03 Feb 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-8" class="internal">Lecture 8</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-30T00:00:00.000Z">30 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-7" class="internal">Lecture 7</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-27T00:00:00.000Z">27 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-6---2.1-Topology" class="internal">Lecture 6 - 2.1 Topology</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-20T00:00:00.000Z">20 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-5" class="internal">Lecture 5</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-16T00:00:00.000Z">16 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" class="internal">Lecture 4 - 1.2 Metric Spaces</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-13T00:00:00.000Z">13 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-3" class="internal">Lecture 3</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-09T00:00:00.000Z">09 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-2" class="internal">Lecture 2</a></h3></div><ul class="tags"></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-01-06T00:00:00.000Z">06 Jan 2025</time></p><div class="desc"><h3><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" class="internal">Lecture 1 - 1.1 Sets and Numbers</a></h3></div><ul class="tags"></ul></div></li></ul></div></div></div><hr/><div class="page-footer"></div></div><div class="right sidebar"></div><footer class><p>Created with <a href="https://quartz.jzhao.xyz/">Quartz v4.4.0</a> © 2025</p><ul><li><a href="https://github.com/jackyzha0/quartz">GitHub</a></li><li><a href="https://discord.gg/cRFFHYye7t">Discord Community</a></li></ul></footer></div></div></body><script 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data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><h1 class="article-title">index</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:26:38.879Z">Mar 01, 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="chapter-1">Chapter 1<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#chapter-1" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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data-folderul="Definitions"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Functions"><button class="folder-button"><span class="folder-title">Functions</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Functions"><li><a href="./Definitions/Functions/Characteristic-Function" data-for="Definitions/Functions/Characteristic-Function">Characteristic Function</a></li><li><a href="./Definitions/Functions/Direct-Product" data-for="Definitions/Functions/Direct-Product">Direct Product</a></li><li><a href="./Definitions/Functions/Inverse-Function" data-for="Definitions/Functions/Inverse-Function">Inverse Function</a></li><li><a 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class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Measure-Theory/Sigma-Algebra"><button class="folder-button"><span class="folder-title">Sigma-Algebra</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Measure-Theory/Sigma-Algebra"><li><a href="./Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable" data-for="Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable">Borel Measurable</a></li><li><a href="./Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets" data-for="Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets">Borel Sets</a></li><li><a href="./Definitions/Measure-Theory/Sigma-Algebra/Measurable" data-for="Definitions/Measure-Theory/Sigma-Algebra/Measurable">Measurable</a></li><li><a href="./Definitions/Measure-Theory/Sigma-Algebra/Measure" data-for="Definitions/Measure-Theory/Sigma-Algebra/Measure">Measure</a></li><li><a href="./Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra" data-for="Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra">Sigma-Algebra</a></li></ul></div></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Metric-Spaces"><button class="folder-button"><span class="folder-title">Metric Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Metric-Spaces"><li><a href="./Definitions/Metric-Spaces/Ball" data-for="Definitions/Metric-Spaces/Ball">Ball</a></li><li><a href="./Definitions/Metric-Spaces/Interior-Point" data-for="Definitions/Metric-Spaces/Interior-Point">Interior Point</a></li><li><a href="./Definitions/Metric-Spaces/Metric-Space" data-for="Definitions/Metric-Spaces/Metric-Space">Metric Space</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Sets"><button class="folder-button"><span class="folder-title">Sets</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Sets"><li><a href="./Definitions/Sets/Complex-Numbers" data-for="Definitions/Sets/Complex-Numbers">Complex Numbers</a></li><li><a href="./Definitions/Sets/Open-Cover" data-for="Definitions/Sets/Open-Cover">Open Cover</a></li><li><a href="./Definitions/Sets/Open-Map" data-for="Definitions/Sets/Open-Map">Open Map</a></li><li><a href="./Definitions/Sets/Open-Sets" data-for="Definitions/Sets/Open-Sets">Open Sets</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Statements"><button class="folder-button"><span class="folder-title">Statements</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Statements"><li><a href="./Definitions/Statements/And" data-for="Definitions/Statements/And">And</a></li><li><a href="./Definitions/Statements/Implies" data-for="Definitions/Statements/Implies">Implies</a></li><li><a href="./Definitions/Statements/Not" data-for="Definitions/Statements/Not">Not</a></li><li><a href="./Definitions/Statements/Or" data-for="Definitions/Statements/Or">Or</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Terminology"><button class="folder-button"><span class="folder-title">Terminology</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Terminology"><li><a href="./Definitions/Terminology/Algebraically-Complete" data-for="Definitions/Terminology/Algebraically-Complete">Algebraically Complete</a></li><li><a href="./Definitions/Terminology/Bijective" data-for="Definitions/Terminology/Bijective">Bijective</a></li><li><a href="./Definitions/Terminology/Bounded" data-for="Definitions/Terminology/Bounded">Bounded</a></li><li><a href="./Definitions/Terminology/Countable" data-for="Definitions/Terminology/Countable">Countable</a></li><li><a href="./Definitions/Terminology/Injective" data-for="Definitions/Terminology/Injective">Injective</a></li><li><a href="./Definitions/Terminology/QED" data-for="Definitions/Terminology/QED">QED</a></li><li><a href="./Definitions/Terminology/Surjective" data-for="Definitions/Terminology/Surjective">Surjective</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces"><button class="folder-button"><span class="folder-title">Topological Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Induced"><button class="folder-button"><span class="folder-title">Induced</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Induced"><li><a href="./Definitions/Topological-Spaces/Induced/Initial-Topology" data-for="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a></li><li><a href="./Definitions/Topological-Spaces/Induced/Product-Topology" data-for="Definitions/Topological-Spaces/Induced/Product-Topology">Product Topology</a></li><li><a href="./Definitions/Topological-Spaces/Induced/Separating-Points" data-for="Definitions/Topological-Spaces/Induced/Separating-Points">Separating Points</a></li><li><a href="./Definitions/Topological-Spaces/Induced/Weakest-Topology" data-for="Definitions/Topological-Spaces/Induced/Weakest-Topology">Weakest Topology</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Terminologies"><button class="folder-button"><span class="folder-title">Terminologies</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Terminologies"><li><a href="./Definitions/Topological-Spaces/Terminologies/Compact" 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href="./Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="./Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="./Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="./Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="./Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="./Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="./Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="./Definitions/Linear-Map" data-for="Definitions/Linear-Map">Linear Map</a></li><li><a href="./Definitions/Nets" data-for="Definitions/Nets">Nets</a></li><li><a href="./Definitions/Norm" data-for="Definitions/Norm">Norm</a></li><li><a href="./Definitions/Number-Field" data-for="Definitions/Number-Field">Number Field</a></li><li><a href="./Definitions/Period-of-a-Fraction" data-for="Definitions/Period-of-a-Fraction">Period of a Fraction</a></li><li><a href="./Definitions/Rational-Cauchy-Sequences" data-for="Definitions/Rational-Cauchy-Sequences">Rational Cauchy Sequences</a></li><li><a href="./Definitions/Subcover" data-for="Definitions/Subcover">Subcover</a></li><li><a href="./Definitions/Subnet" data-for="Definitions/Subnet">Subnet</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="./Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 10.57.31.excalidraw</a></li><li><a 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href="./Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="./Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a 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href="./Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="./Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="./Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="./Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="./Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="./Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="./Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="./Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="./Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="./Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="./Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="./Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><h1 class="article-title">index</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T13:37:40.378Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="chapter-1">Chapter 1<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#chapter-1" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
 <h2 id="11-sets-and-numbers">1.1 Sets and Numbers<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#11-sets-and-numbers" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
 <ul>
 <li><a href="./Lectures/Lecture-1---1.1-Sets-and-Numbers" class="internal alias" data-slug="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a> (complimentary written notes: <a href="./rnote/ACIT4330-2025-01-06-Lecture-1.rnote" class="internal alias" data-slug="rnote/ACIT4330-2025-01-06-Lecture-1.rnote">ACIT4330-2025-01-06-Lecture 1.rnote</a>)</li>
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+    <guid>https://quartz.jzhao.xyz/Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw</guid>
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data-for="Definitions/Statements/Or">Or</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Terminology"><button class="folder-button"><span class="folder-title">Terminology</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Terminology"><li><a href="../Definitions/Terminology/Algebraically-Complete" data-for="Definitions/Terminology/Algebraically-Complete">Algebraically Complete</a></li><li><a href="../Definitions/Terminology/Bijective" data-for="Definitions/Terminology/Bijective">Bijective</a></li><li><a href="../Definitions/Terminology/Bounded" data-for="Definitions/Terminology/Bounded">Bounded</a></li><li><a href="../Definitions/Terminology/Countable" data-for="Definitions/Terminology/Countable">Countable</a></li><li><a href="../Definitions/Terminology/Injective" data-for="Definitions/Terminology/Injective">Injective</a></li><li><a href="../Definitions/Terminology/QED" data-for="Definitions/Terminology/QED">QED</a></li><li><a href="../Definitions/Terminology/Surjective" data-for="Definitions/Terminology/Surjective">Surjective</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces"><button class="folder-button"><span class="folder-title">Topological Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Induced"><button class="folder-button"><span class="folder-title">Induced</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Induced"><li><a href="../Definitions/Topological-Spaces/Induced/Initial-Topology" data-for="Definitions/Topological-Spaces/Induced/Initial-Topology">Initial Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Product-Topology" data-for="Definitions/Topological-Spaces/Induced/Product-Topology">Product Topology</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Separating-Points" data-for="Definitions/Topological-Spaces/Induced/Separating-Points">Separating Points</a></li><li><a href="../Definitions/Topological-Spaces/Induced/Weakest-Topology" data-for="Definitions/Topological-Spaces/Induced/Weakest-Topology">Weakest Topology</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Topological-Spaces/Terminologies"><button class="folder-button"><span class="folder-title">Terminologies</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Topological-Spaces/Terminologies"><li><a href="../Definitions/Topological-Spaces/Terminologies/Compact" data-for="Definitions/Topological-Spaces/Terminologies/Compact">Compact</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected" data-for="Definitions/Topological-Spaces/Terminologies/Connected">Connected</a></li><li><a href="../Definitions/Topological-Spaces/Terminologies/Connected-Component" data-for="Definitions/Topological-Spaces/Terminologies/Connected-Component">Connected Component</a></li></ul></div></li><li><a href="../Definitions/Topological-Spaces/Continuous" data-for="Definitions/Topological-Spaces/Continuous">Continuous</a></li><li><a href="../Definitions/Topological-Spaces/Hausdorff" data-for="Definitions/Topological-Spaces/Hausdorff">Hausdorff</a></li><li><a href="../Definitions/Topological-Spaces/Topological-Space" data-for="Definitions/Topological-Spaces/Topological-Space">Topological Space</a></li><li><a href="../Definitions/Topological-Spaces/Topology" data-for="Definitions/Topological-Spaces/Topology">Topology</a></li><li><a href="../Definitions/Topological-Spaces/Tychonoff-Theorem" data-for="Definitions/Topological-Spaces/Tychonoff-Theorem">Tychonoff Theorem</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Definitions/Vector-Spaces"><button class="folder-button"><span class="folder-title">Vector Spaces</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Vector-Spaces"><li><a href="../Definitions/Vector-Spaces/Complex-Vector-Space" data-for="Definitions/Vector-Spaces/Complex-Vector-Space">Complex Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Linear-Basis" data-for="Definitions/Vector-Spaces/Linear-Basis">Linear Basis</a></li><li><a href="../Definitions/Vector-Spaces/Normed-Vector-Space" data-for="Definitions/Vector-Spaces/Normed-Vector-Space">Normed Vector Space</a></li><li><a href="../Definitions/Vector-Spaces/Properties-of-a-Vector-Space" data-for="Definitions/Vector-Spaces/Properties-of-a-Vector-Space">Properties of a Vector Space</a></li></ul></div></li><li><a href="../Definitions/Cauchy-Sequence" data-for="Definitions/Cauchy-Sequence">Cauchy Sequence</a></li><li><a href="../Definitions/Cauchy-Schwarz-Inequality" data-for="Definitions/Cauchy-Schwarz-Inequality">Cauchy-Schwarz Inequality</a></li><li><a href="../Definitions/Hilbert-Spaces" data-for="Definitions/Hilbert-Spaces">Hilbert Spaces</a></li><li><a href="../Definitions/Inner-Product" data-for="Definitions/Inner-Product">Inner Product</a></li><li><a href="../Definitions/Least-Upper-Bound-Property" data-for="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></li><li><a href="../Definitions/Linear-Map" data-for="Definitions/Linear-Map">Linear Map</a></li><li><a href="../Definitions/Nets" data-for="Definitions/Nets">Nets</a></li><li><a href="../Definitions/Norm" data-for="Definitions/Norm">Norm</a></li><li><a href="../Definitions/Number-Field" data-for="Definitions/Number-Field">Number Field</a></li><li><a href="../Definitions/Period-of-a-Fraction" data-for="Definitions/Period-of-a-Fraction">Period of a Fraction</a></li><li><a href="../Definitions/Rational-Cauchy-Sequences" data-for="Definitions/Rational-Cauchy-Sequences">Rational Cauchy Sequences</a></li><li><a href="../Definitions/Subcover" data-for="Definitions/Subcover">Subcover</a></li><li><a href="../Definitions/Subnet" data-for="Definitions/Subnet">Subnet</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw"><button class="folder-button"><span class="folder-title">Excalidraw</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw"><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-11"><button class="folder-button"><span class="folder-title">Lecture 11</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-11"><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw">Drawing 2025-02-13 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12.07.00.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw">Drawing 2025-02-13 14.01.44.excalidraw</a></li><li><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" data-for="Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw">Drawing 2025-02-13 14.06.25.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-12"><button class="folder-button"><span class="folder-title">Lecture 12</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-12"><li><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw" data-for="Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw">Drawing 2025-02-24 11.58.37.excalidraw</a></li><li><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw" data-for="Excalidraw/Lecture-12/Drawing-2025-02-24-12.47.32.excalidraw">Drawing 2025-02-24 12.47.32.excalidraw</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Excalidraw/Lecture-13"><button class="folder-button"><span class="folder-title">Lecture 13</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Excalidraw/Lecture-13"><li><a href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" data-for="Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw">Drawing 2025-02-27 13.19.24.excalidraw</a></li></ul></div></li></ul></div></li><li><div class="folder-outer "><ul style="padding-left:0;" class="content" data-folderul></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" 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href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../tags/">tags</a></div></nav><h1 class="article-title">Tag Index</h1></div></div><div class="popover-hint"><article class><p></p></article><p>Found 1 total tags.</p><div><div><h2><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></h2><div class="page-listing"><p>11 items with this tag. <span>Showing first 10 tags.</span></p><ul class="section-ul"><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.386Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw" class="internal">Drawing 2025-02-13 11.26.08.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.386Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-13/Drawing-2025-02-27-13.19.24.excalidraw" class="internal">Drawing 2025-02-27 13.19.24.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.385Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw" class="internal">Drawing 2025-02-13 10.57.31.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.385Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.01.57.excalidraw" class="internal">Drawing 2025-02-13 12.01.57.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.384Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.06.25.excalidraw" class="internal">Drawing 2025-02-13 14.06.25.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.383Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-12.07.00.excalidraw" class="internal">Drawing 2025-02-13 12.07.00.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.382Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.02.27.excalidraw" class="internal">Drawing 2025-02-13 11.02.27.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.382Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-11.47.33.excalidraw" class="internal">Drawing 2025-02-13 11.47.33.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.382Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-11/Drawing-2025-02-13-14.01.44.excalidraw" class="internal">Drawing 2025-02-13 14.01.44.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li><li class="section-li"><div class="section"><p class="meta"><time datetime="2025-03-01T13:37:40.380Z">01 Mar 2025</time></p><div class="desc"><h3><a href="../Excalidraw/Lecture-12/Drawing-2025-02-24-11.58.37.excalidraw" class="internal">Drawing 2025-02-24 11.58.37.excalidraw</a></h3></div><ul class="tags"><li><a class="internal tag-link" href="../tags/excalidraw">excalidraw</a></li></ul></div></li></ul></div></div></div></div><hr/><div class="page-footer"></div></div><div class="right sidebar"></div><footer class><p>Created with <a href="https://quartz.jzhao.xyz/">Quartz v4.4.0</a> © 2025</p><ul><li><a href="https://github.com/jackyzha0/quartz">GitHub</a></li><li><a href="https://discord.gg/cRFFHYye7t">Discord Community</a></li></ul></footer></div></div></body><script type="application/javascript">function c(){let t=this.parentElement;t.classList.toggle("is-collapsed");let l=t.classList.contains("is-collapsed")?this.scrollHeight:t.scrollHeight;t.style.maxHeight=l+"px";let o=t,e=t.parentElement;for(;e;){if(!e.classList.contains("callout"))return;let n=e.classList.contains("is-collapsed")?e.scrollHeight:e.scrollHeight+o.scrollHeight;e.style.maxHeight=n+"px",o=e,e=e.parentElement}}function i(){let t=document.getElementsByClassName("callout is-collapsible");for(let s of t){let l=s.firstElementChild;if(l){l.addEventListener("click",c),window.addCleanup(()=>l.removeEventListener("click",c));let e=s.classList.contains("is-collapsed")?l.scrollHeight:s.scrollHeight;s.style.maxHeight=e+"px"}}}document.addEventListener("nav",i);window.addEventListener("resize",i);
 </script><script src="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/contrib/copy-tex.min.js" type="application/javascript"></script><script type="application/javascript">
         const socket = new WebSocket('ws://localhost:3001')
         // reload(true) ensures resources like images and scripts are fetched again in firefox
diff --git a/quartz.config.ts b/quartz.config.ts
index 5cc001b8..d7470b30 100644
--- a/quartz.config.ts
+++ b/quartz.config.ts
@@ -8,14 +8,14 @@ import * as Plugin from "./quartz/plugins"
  */
 const config: QuartzConfig = {
   configuration: {
-    pageTitle: "🪴 Quartz 4.0",
+    pageTitle: "ACIT4330 Lecture Notes",
     pageTitleSuffix: "",
     enableSPA: true,
     enablePopovers: true,
     analytics: {
       provider: "plausible",
     },
-    locale: "en-US",
+    locale: "en-GB",
     baseUrl: "quartz.jzhao.xyz",
     ignorePatterns: ["private", "templates", ".obsidian"],
     defaultDateType: "created",
diff --git a/quartz/.quartz-cache/transpiled-build.mjs b/quartz/.quartz-cache/transpiled-build.mjs
index 0ddb0735..19c7bb84 100644
--- a/quartz/.quartz-cache/transpiled-build.mjs
+++ b/quartz/.quartz-cache/transpiled-build.mjs
@@ -3160,6 +3160,6 @@ a:hover .popover,
         const socket = new WebSocket('${wsUrl}')
         // reload(true) ensures resources like images and scripts are fetched again in firefox
         socket.addEventListener('message', () => document.location.reload(true))
-      `})}return staticResources}__name(getStaticResourcesFromPlugins,"getStaticResourcesFromPlugins");async function emitContent(ctx,content){let{argv,cfg}=ctx,perf=new PerfTimer,log=new QuartzLogger(ctx.argv.verbose);log.start("Emitting output files");let emittedFiles=0,staticResources=getStaticResourcesFromPlugins(ctx);for(let emitter of cfg.plugins.emitters)try{let emitted=await emitter.emit(ctx,content,staticResources);if(emittedFiles+=emitted.length,ctx.argv.verbose)for(let file of emitted)console.log(`[emit:${emitter.name}] ${file}`)}catch(err){trace(`Failed to emit from plugin \`${emitter.name}\``,err)}log.end(`Emitted ${emittedFiles} files to \`${argv.output}\` in ${perf.timeSince()}`)}__name(emitContent,"emitContent");var config={configuration:{pageTitle:"\u{1FAB4} Quartz 4.0",pageTitleSuffix:"",enableSPA:!0,enablePopovers:!0,analytics:{provider:"plausible"},locale:"en-US",baseUrl:"quartz.jzhao.xyz",ignorePatterns:["private","templates",".obsidian"],defaultDateType:"created",generateSocialImages:!1,theme:{fontOrigin:"googleFonts",cdnCaching:!0,typography:{header:"Schibsted Grotesk",body:"Source Sans Pro",code:"IBM Plex Mono"},colors:{lightMode:{light:"#faf8f8",lightgray:"#e5e5e5",gray:"#b8b8b8",darkgray:"#4e4e4e",dark:"#2b2b2b",secondary:"#284b63",tertiary:"#84a59d",highlight:"rgba(143, 159, 169, 0.15)",textHighlight:"#fff23688"},darkMode:{light:"#161618",lightgray:"#393639",gray:"#646464",darkgray:"#d4d4d4",dark:"#ebebec",secondary:"#7b97aa",tertiary:"#84a59d",highlight:"rgba(143, 159, 169, 0.15)",textHighlight:"#b3aa0288"}}}},plugins:{transformers:[FrontMatter(),CreatedModifiedDate({priority:["frontmatter","filesystem"]}),SyntaxHighlighting({theme:{light:"github-light",dark:"github-dark"},keepBackground:!1}),ObsidianFlavoredMarkdown({enableInHtmlEmbed:!1}),GitHubFlavoredMarkdown(),TableOfContents(),CrawlLinks({markdownLinkResolution:"shortest"}),Description(),Latex({renderEngine:"katex"}),HardLineBreaks()],filters:[RemoveDrafts()],emitters:[AliasRedirects(),ComponentResources(),ContentPage(),FolderPage(),TagPage(),ContentIndex({enableSiteMap:!0,enableRSS:!0}),Assets(),Static(),NotFoundPage()]}},quartz_config_default=config;import chokidar from"chokidar";import fs6 from"fs";import{fileURLToPath}from"url";var options={retrieveSourceMap(source){if(source.includes(".quartz-cache")){let realSource=fileURLToPath(source.split("?",2)[0]+".map");return{map:fs6.readFileSync(realSource,"utf8")}}else return null}};sourceMapSupport.install(options);function newBuildId(){return Math.random().toString(36).substring(2,8)}__name(newBuildId,"newBuildId");async function buildQuartz(argv,mut,clientRefresh){let ctx={buildId:newBuildId(),argv,cfg:quartz_config_default,allSlugs:[]},perf=new PerfTimer,output=argv.output,pluginCount=Object.values(quartz_config_default.plugins).flat().length,pluginNames=__name(key=>quartz_config_default.plugins[key].map(plugin=>plugin.name),"pluginNames");argv.verbose&&(console.log(`Loaded ${pluginCount} plugins`),console.log(`  Transformers: ${pluginNames("transformers").join(", ")}`),console.log(`  Filters: ${pluginNames("filters").join(", ")}`),console.log(`  Emitters: ${pluginNames("emitters").join(", ")}`));let release=await mut.acquire();perf.addEvent("clean"),await rimraf(path12.join(output,"*"),{glob:!0}),console.log(`Cleaned output directory \`${output}\` in ${perf.timeSince("clean")}`),perf.addEvent("glob");let allFiles=await glob("**/*.*",argv.directory,quartz_config_default.configuration.ignorePatterns),fps=allFiles.filter(fp=>fp.endsWith(".md")).sort();console.log(`Found ${fps.length} input files from \`${argv.directory}\` in ${perf.timeSince("glob")}`);let filePaths=fps.map(fp=>joinSegments(argv.directory,fp));ctx.allSlugs=allFiles.map(fp=>slugifyFilePath(fp));let parsedFiles=await parseMarkdown(ctx,filePaths),filteredContent=filterContent(ctx,parsedFiles),dependencies={};if(argv.fastRebuild){let staticResources=getStaticResourcesFromPlugins(ctx);for(let emitter of quartz_config_default.plugins.emitters)dependencies[emitter.name]=await emitter.getDependencyGraph?.(ctx,filteredContent,staticResources)??null}if(await emitContent(ctx,filteredContent),console.log(chalk6.green(`Done processing ${fps.length} files in ${perf.timeSince()}`)),release(),argv.serve)return startServing(ctx,mut,parsedFiles,clientRefresh,dependencies)}__name(buildQuartz,"buildQuartz");async function startServing(ctx,mut,initialContent,clientRefresh,dependencies){let{argv}=ctx,contentMap=new Map;for(let content of initialContent){let[_tree,vfile]=content;contentMap.set(vfile.data.filePath,content)}let buildData={ctx,mut,dependencies,contentMap,ignored:await isGitIgnored(),initialSlugs:ctx.allSlugs,toRebuild:new Set,toRemove:new Set,trackedAssets:new Set,lastBuildMs:0},watcher=chokidar.watch(".",{persistent:!0,cwd:argv.directory,ignoreInitial:!0}),buildFromEntry=argv.fastRebuild?partialRebuildFromEntrypoint:rebuildFromEntrypoint;return watcher.on("add",fp=>buildFromEntry(fp,"add",clientRefresh,buildData)).on("change",fp=>buildFromEntry(fp,"change",clientRefresh,buildData)).on("unlink",fp=>buildFromEntry(fp,"delete",clientRefresh,buildData)),async()=>{await watcher.close()}}__name(startServing,"startServing");async function partialRebuildFromEntrypoint(filepath,action,clientRefresh,buildData){let{ctx,ignored,dependencies,contentMap,mut,toRemove}=buildData,{argv,cfg}=ctx;if(ignored(filepath))return;let buildId=newBuildId();ctx.buildId=buildId,buildData.lastBuildMs=new Date().getTime();let release=await mut.acquire();if(ctx.buildId!==buildId){release();return}let perf=new PerfTimer;console.log(chalk6.yellow("Detected change, rebuilding..."));let fp=joinSegments(argv.directory,toPosixPath(filepath)),staticResources=getStaticResourcesFromPlugins(ctx),processedFiles=[];switch(action){case"add":processedFiles=await parseMarkdown(ctx,[fp]),processedFiles.forEach(([tree,vfile])=>contentMap.set(vfile.data.filePath,[tree,vfile]));for(let emitter of cfg.plugins.emitters){let emitterGraph=await emitter.getDependencyGraph?.(ctx,processedFiles,staticResources)??null;if(emitterGraph){let existingGraph=dependencies[emitter.name];existingGraph!==null?existingGraph.mergeGraph(emitterGraph):dependencies[emitter.name]=emitterGraph}}break;case"change":if(processedFiles=await parseMarkdown(ctx,[fp]),processedFiles.forEach(([tree,vfile])=>contentMap.set(vfile.data.filePath,[tree,vfile])),path12.extname(fp)===".md")for(let emitter of cfg.plugins.emitters){let emitterGraph=await emitter.getDependencyGraph?.(ctx,processedFiles,staticResources)??null;emitterGraph?.hasNode(fp)&&dependencies[emitter.name]?.updateIncomingEdgesForNode(emitterGraph,fp)}break;case"delete":toRemove.add(fp);break}argv.verbose&&console.log(`Updated dependency graphs in ${perf.timeSince()}`),perf.addEvent("rebuild");let emittedFiles=0;for(let emitter of cfg.plugins.emitters){let depGraph=dependencies[emitter.name];if(depGraph===null){argv.verbose&&console.log(`Emitter ${emitter.name} doesn't define a dependency graph. Calling it with all files...`);let files=[...contentMap.values()].filter(([_node,vfile])=>!toRemove.has(vfile.data.filePath)),emittedFps=await emitter.emit(ctx,files,staticResources);if(ctx.argv.verbose)for(let file of emittedFps)console.log(`[emit:${emitter.name}] ${file}`);emittedFiles+=emittedFps.length;continue}if(depGraph.hasNode(fp)){let upstreamContent=[...depGraph.getLeafNodeAncestors(fp)].filter(file=>contentMap.has(file)).filter(file=>!toRemove.has(file)).map(file=>contentMap.get(file)),emittedFps=await emitter.emit(ctx,upstreamContent,staticResources);if(ctx.argv.verbose)for(let file of emittedFps)console.log(`[emit:${emitter.name}] ${file}`);emittedFiles+=emittedFps.length}}console.log(`Emitted ${emittedFiles} files to \`${argv.output}\` in ${perf.timeSince("rebuild")}`);let destinationsToDelete=new Set;for(let file of toRemove)contentMap.delete(file),Object.values(dependencies).forEach(depGraph=>{depGraph?.removeNode(file),depGraph?.removeOrphanNodes()?.forEach(node=>{node.startsWith(argv.output)&&destinationsToDelete.add(node)})});await rimraf([...destinationsToDelete]),console.log(chalk6.green(`Done rebuilding in ${perf.timeSince()}`)),toRemove.clear(),release(),clientRefresh()}__name(partialRebuildFromEntrypoint,"partialRebuildFromEntrypoint");async function rebuildFromEntrypoint(fp,action,clientRefresh,buildData){let{ctx,ignored,mut,initialSlugs,contentMap,toRebuild,toRemove,trackedAssets}=buildData,{argv}=ctx;if(ignored(fp))return;fp=toPosixPath(fp);let filePath=joinSegments(argv.directory,fp);if(path12.extname(fp)!==".md"){action==="add"||action==="change"?trackedAssets.add(filePath):action==="delete"&&trackedAssets.delete(filePath),clientRefresh();return}action==="add"||action==="change"?toRebuild.add(filePath):action==="delete"&&toRemove.add(filePath);let buildId=newBuildId();ctx.buildId=buildId,buildData.lastBuildMs=new Date().getTime();let release=await mut.acquire();if(ctx.buildId!==buildId){release();return}let perf=new PerfTimer;console.log(chalk6.yellow("Detected change, rebuilding..."));try{let filesToRebuild=[...toRebuild].filter(fp2=>!toRemove.has(fp2)),parsedContent=await parseMarkdown(ctx,filesToRebuild);for(let content of parsedContent){let[_tree,vfile]=content;contentMap.set(vfile.data.filePath,content)}for(let fp2 of toRemove)contentMap.delete(fp2);let parsedFiles=[...contentMap.values()],filteredContent=filterContent(ctx,parsedFiles),trackedSlugs=[...new Set([...contentMap.keys(),...toRebuild,...trackedAssets])].filter(fp2=>!toRemove.has(fp2)).map(fp2=>slugifyFilePath(path12.posix.relative(argv.directory,fp2)));ctx.allSlugs=[...new Set([...initialSlugs,...trackedSlugs])],await rimraf(path12.join(argv.output,".*"),{glob:!0}),await emitContent(ctx,filteredContent),console.log(chalk6.green(`Done rebuilding in ${perf.timeSince()}`))}catch(err){console.log(chalk6.yellow("Rebuild failed. Waiting on a change to fix the error...")),argv.verbose&&console.log(chalk6.red(err))}clientRefresh(),toRebuild.clear(),toRemove.clear(),release()}__name(rebuildFromEntrypoint,"rebuildFromEntrypoint");var build_default=__name(async(argv,mut,clientRefresh)=>{try{return await buildQuartz(argv,mut,clientRefresh)}catch(err){trace(`
+      `})}return staticResources}__name(getStaticResourcesFromPlugins,"getStaticResourcesFromPlugins");async function emitContent(ctx,content){let{argv,cfg}=ctx,perf=new PerfTimer,log=new QuartzLogger(ctx.argv.verbose);log.start("Emitting output files");let emittedFiles=0,staticResources=getStaticResourcesFromPlugins(ctx);for(let emitter of cfg.plugins.emitters)try{let emitted=await emitter.emit(ctx,content,staticResources);if(emittedFiles+=emitted.length,ctx.argv.verbose)for(let file of emitted)console.log(`[emit:${emitter.name}] ${file}`)}catch(err){trace(`Failed to emit from plugin \`${emitter.name}\``,err)}log.end(`Emitted ${emittedFiles} files to \`${argv.output}\` in ${perf.timeSince()}`)}__name(emitContent,"emitContent");var config={configuration:{pageTitle:"ACIT4330 Lecture Notes",pageTitleSuffix:"",enableSPA:!0,enablePopovers:!0,analytics:{provider:"plausible"},locale:"en-GB",baseUrl:"quartz.jzhao.xyz",ignorePatterns:["private","templates",".obsidian"],defaultDateType:"created",generateSocialImages:!1,theme:{fontOrigin:"googleFonts",cdnCaching:!0,typography:{header:"Schibsted Grotesk",body:"Source Sans Pro",code:"IBM Plex Mono"},colors:{lightMode:{light:"#faf8f8",lightgray:"#e5e5e5",gray:"#b8b8b8",darkgray:"#4e4e4e",dark:"#2b2b2b",secondary:"#284b63",tertiary:"#84a59d",highlight:"rgba(143, 159, 169, 0.15)",textHighlight:"#fff23688"},darkMode:{light:"#161618",lightgray:"#393639",gray:"#646464",darkgray:"#d4d4d4",dark:"#ebebec",secondary:"#7b97aa",tertiary:"#84a59d",highlight:"rgba(143, 159, 169, 0.15)",textHighlight:"#b3aa0288"}}}},plugins:{transformers:[FrontMatter(),CreatedModifiedDate({priority:["frontmatter","filesystem"]}),SyntaxHighlighting({theme:{light:"github-light",dark:"github-dark"},keepBackground:!1}),ObsidianFlavoredMarkdown({enableInHtmlEmbed:!1}),GitHubFlavoredMarkdown(),TableOfContents(),CrawlLinks({markdownLinkResolution:"shortest"}),Description(),Latex({renderEngine:"katex"}),HardLineBreaks()],filters:[RemoveDrafts()],emitters:[AliasRedirects(),ComponentResources(),ContentPage(),FolderPage(),TagPage(),ContentIndex({enableSiteMap:!0,enableRSS:!0}),Assets(),Static(),NotFoundPage()]}},quartz_config_default=config;import chokidar from"chokidar";import fs6 from"fs";import{fileURLToPath}from"url";var options={retrieveSourceMap(source){if(source.includes(".quartz-cache")){let realSource=fileURLToPath(source.split("?",2)[0]+".map");return{map:fs6.readFileSync(realSource,"utf8")}}else return null}};sourceMapSupport.install(options);function newBuildId(){return Math.random().toString(36).substring(2,8)}__name(newBuildId,"newBuildId");async function buildQuartz(argv,mut,clientRefresh){let ctx={buildId:newBuildId(),argv,cfg:quartz_config_default,allSlugs:[]},perf=new PerfTimer,output=argv.output,pluginCount=Object.values(quartz_config_default.plugins).flat().length,pluginNames=__name(key=>quartz_config_default.plugins[key].map(plugin=>plugin.name),"pluginNames");argv.verbose&&(console.log(`Loaded ${pluginCount} plugins`),console.log(`  Transformers: ${pluginNames("transformers").join(", ")}`),console.log(`  Filters: ${pluginNames("filters").join(", ")}`),console.log(`  Emitters: ${pluginNames("emitters").join(", ")}`));let release=await mut.acquire();perf.addEvent("clean"),await rimraf(path12.join(output,"*"),{glob:!0}),console.log(`Cleaned output directory \`${output}\` in ${perf.timeSince("clean")}`),perf.addEvent("glob");let allFiles=await glob("**/*.*",argv.directory,quartz_config_default.configuration.ignorePatterns),fps=allFiles.filter(fp=>fp.endsWith(".md")).sort();console.log(`Found ${fps.length} input files from \`${argv.directory}\` in ${perf.timeSince("glob")}`);let filePaths=fps.map(fp=>joinSegments(argv.directory,fp));ctx.allSlugs=allFiles.map(fp=>slugifyFilePath(fp));let parsedFiles=await parseMarkdown(ctx,filePaths),filteredContent=filterContent(ctx,parsedFiles),dependencies={};if(argv.fastRebuild){let staticResources=getStaticResourcesFromPlugins(ctx);for(let emitter of quartz_config_default.plugins.emitters)dependencies[emitter.name]=await emitter.getDependencyGraph?.(ctx,filteredContent,staticResources)??null}if(await emitContent(ctx,filteredContent),console.log(chalk6.green(`Done processing ${fps.length} files in ${perf.timeSince()}`)),release(),argv.serve)return startServing(ctx,mut,parsedFiles,clientRefresh,dependencies)}__name(buildQuartz,"buildQuartz");async function startServing(ctx,mut,initialContent,clientRefresh,dependencies){let{argv}=ctx,contentMap=new Map;for(let content of initialContent){let[_tree,vfile]=content;contentMap.set(vfile.data.filePath,content)}let buildData={ctx,mut,dependencies,contentMap,ignored:await isGitIgnored(),initialSlugs:ctx.allSlugs,toRebuild:new Set,toRemove:new Set,trackedAssets:new Set,lastBuildMs:0},watcher=chokidar.watch(".",{persistent:!0,cwd:argv.directory,ignoreInitial:!0}),buildFromEntry=argv.fastRebuild?partialRebuildFromEntrypoint:rebuildFromEntrypoint;return watcher.on("add",fp=>buildFromEntry(fp,"add",clientRefresh,buildData)).on("change",fp=>buildFromEntry(fp,"change",clientRefresh,buildData)).on("unlink",fp=>buildFromEntry(fp,"delete",clientRefresh,buildData)),async()=>{await watcher.close()}}__name(startServing,"startServing");async function partialRebuildFromEntrypoint(filepath,action,clientRefresh,buildData){let{ctx,ignored,dependencies,contentMap,mut,toRemove}=buildData,{argv,cfg}=ctx;if(ignored(filepath))return;let buildId=newBuildId();ctx.buildId=buildId,buildData.lastBuildMs=new Date().getTime();let release=await mut.acquire();if(ctx.buildId!==buildId){release();return}let perf=new PerfTimer;console.log(chalk6.yellow("Detected change, rebuilding..."));let fp=joinSegments(argv.directory,toPosixPath(filepath)),staticResources=getStaticResourcesFromPlugins(ctx),processedFiles=[];switch(action){case"add":processedFiles=await parseMarkdown(ctx,[fp]),processedFiles.forEach(([tree,vfile])=>contentMap.set(vfile.data.filePath,[tree,vfile]));for(let emitter of cfg.plugins.emitters){let emitterGraph=await emitter.getDependencyGraph?.(ctx,processedFiles,staticResources)??null;if(emitterGraph){let existingGraph=dependencies[emitter.name];existingGraph!==null?existingGraph.mergeGraph(emitterGraph):dependencies[emitter.name]=emitterGraph}}break;case"change":if(processedFiles=await parseMarkdown(ctx,[fp]),processedFiles.forEach(([tree,vfile])=>contentMap.set(vfile.data.filePath,[tree,vfile])),path12.extname(fp)===".md")for(let emitter of cfg.plugins.emitters){let emitterGraph=await emitter.getDependencyGraph?.(ctx,processedFiles,staticResources)??null;emitterGraph?.hasNode(fp)&&dependencies[emitter.name]?.updateIncomingEdgesForNode(emitterGraph,fp)}break;case"delete":toRemove.add(fp);break}argv.verbose&&console.log(`Updated dependency graphs in ${perf.timeSince()}`),perf.addEvent("rebuild");let emittedFiles=0;for(let emitter of cfg.plugins.emitters){let depGraph=dependencies[emitter.name];if(depGraph===null){argv.verbose&&console.log(`Emitter ${emitter.name} doesn't define a dependency graph. Calling it with all files...`);let files=[...contentMap.values()].filter(([_node,vfile])=>!toRemove.has(vfile.data.filePath)),emittedFps=await emitter.emit(ctx,files,staticResources);if(ctx.argv.verbose)for(let file of emittedFps)console.log(`[emit:${emitter.name}] ${file}`);emittedFiles+=emittedFps.length;continue}if(depGraph.hasNode(fp)){let upstreamContent=[...depGraph.getLeafNodeAncestors(fp)].filter(file=>contentMap.has(file)).filter(file=>!toRemove.has(file)).map(file=>contentMap.get(file)),emittedFps=await emitter.emit(ctx,upstreamContent,staticResources);if(ctx.argv.verbose)for(let file of emittedFps)console.log(`[emit:${emitter.name}] ${file}`);emittedFiles+=emittedFps.length}}console.log(`Emitted ${emittedFiles} files to \`${argv.output}\` in ${perf.timeSince("rebuild")}`);let destinationsToDelete=new Set;for(let file of toRemove)contentMap.delete(file),Object.values(dependencies).forEach(depGraph=>{depGraph?.removeNode(file),depGraph?.removeOrphanNodes()?.forEach(node=>{node.startsWith(argv.output)&&destinationsToDelete.add(node)})});await rimraf([...destinationsToDelete]),console.log(chalk6.green(`Done rebuilding in ${perf.timeSince()}`)),toRemove.clear(),release(),clientRefresh()}__name(partialRebuildFromEntrypoint,"partialRebuildFromEntrypoint");async function rebuildFromEntrypoint(fp,action,clientRefresh,buildData){let{ctx,ignored,mut,initialSlugs,contentMap,toRebuild,toRemove,trackedAssets}=buildData,{argv}=ctx;if(ignored(fp))return;fp=toPosixPath(fp);let filePath=joinSegments(argv.directory,fp);if(path12.extname(fp)!==".md"){action==="add"||action==="change"?trackedAssets.add(filePath):action==="delete"&&trackedAssets.delete(filePath),clientRefresh();return}action==="add"||action==="change"?toRebuild.add(filePath):action==="delete"&&toRemove.add(filePath);let buildId=newBuildId();ctx.buildId=buildId,buildData.lastBuildMs=new Date().getTime();let release=await mut.acquire();if(ctx.buildId!==buildId){release();return}let perf=new PerfTimer;console.log(chalk6.yellow("Detected change, rebuilding..."));try{let filesToRebuild=[...toRebuild].filter(fp2=>!toRemove.has(fp2)),parsedContent=await parseMarkdown(ctx,filesToRebuild);for(let content of parsedContent){let[_tree,vfile]=content;contentMap.set(vfile.data.filePath,content)}for(let fp2 of toRemove)contentMap.delete(fp2);let parsedFiles=[...contentMap.values()],filteredContent=filterContent(ctx,parsedFiles),trackedSlugs=[...new Set([...contentMap.keys(),...toRebuild,...trackedAssets])].filter(fp2=>!toRemove.has(fp2)).map(fp2=>slugifyFilePath(path12.posix.relative(argv.directory,fp2)));ctx.allSlugs=[...new Set([...initialSlugs,...trackedSlugs])],await rimraf(path12.join(argv.output,".*"),{glob:!0}),await emitContent(ctx,filteredContent),console.log(chalk6.green(`Done rebuilding in ${perf.timeSince()}`))}catch(err){console.log(chalk6.yellow("Rebuild failed. Waiting on a change to fix the error...")),argv.verbose&&console.log(chalk6.red(err))}clientRefresh(),toRebuild.clear(),toRemove.clear(),release()}__name(rebuildFromEntrypoint,"rebuildFromEntrypoint");var build_default=__name(async(argv,mut,clientRefresh)=>{try{return await buildQuartz(argv,mut,clientRefresh)}catch(err){trace(`
 Exiting Quartz due to a fatal error`,err)}},"default");export{build_default as default};
 //# sourceMappingURL=transpiled-build.mjs.map
diff --git a/quartz/.quartz-cache/transpiled-build.mjs.map b/quartz/.quartz-cache/transpiled-build.mjs.map
index ebf7468b..ef3a876e 100644
--- a/quartz/.quartz-cache/transpiled-build.mjs.map
+++ b/quartz/.quartz-cache/transpiled-build.mjs.map
@@ -1,6 +1,6 @@
 {
   "version": 3,
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