ACIT4330-Page/public/Lectures/Lecture-6---2.1-Topology.html
Anthony Berg 07737d3368
Some checks are pending
Build and Test / build-and-test (macos-latest) (push) Waiting to run
Build and Test / build-and-test (ubuntu-latest) (push) Waiting to run
Build and Test / build-and-test (windows-latest) (push) Waiting to run
Build and Test / publish-tag (push) Waiting to run
Docker build & push image / build (push) Waiting to run
Quartz sync: Mar 1, 2025, 2:40 PM
2025-03-01 14:40:30 +01:00

66 lines
54 KiB
HTML
Raw Blame History

This file contains invisible Unicode characters

This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE html>
<html lang="en"><head><title>Lecture 6 - 2.1 Topology</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 6 - 2.1 Topology"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 6 - 2.1 Topology"/><meta name="twitter:description" content="Open Sets The concept of balls is required to understand what an open sets are. Definition The definition can be found in Open Sets. In other words ( ”):” ), for any open \overbrace{A}^{\in x} \exists N such that x_{n} \in A \; \forall n \gt N."/><meta property="og:description" content="Open Sets The concept of balls is required to understand what an open sets are. Definition The definition can be found in Open Sets. In other words ( ”):” ), for any open \overbrace{A}^{\in x} \exists N such that x_{n} \in A \; \forall n \gt N."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Open Sets The concept of balls is required to understand what an open sets are. Definition The definition can be found in Open Sets. In other words ( ”):” ), for any open \overbrace{A}^{\in x} \exists N such that x_{n} \in A \; \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/Lectures/Lecture-6---2.1-Topology"/><meta property="twitter:url" content="https://quartz.jzhao.xyz/Lectures/Lecture-6---2.1-Topology"/><link rel="icon" href="../static/icon.png"/><meta name="description" content="Open Sets The concept of balls is required to understand what an open sets are. Definition The definition can be found in Open Sets. In other words ( ”):” ), for any open \overbrace{A}^{\in x} \exists N such that x_{n} \in A \; \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="Lectures/Lecture-6---2.1-Topology"><div id="quartz-root" class="page"><div id="quartz-body"><div class="left sidebar"><h2 class="page-title"><a href="..">ACIT4330 Lecture Notes</a></h2><div class="spacer mobile-only"></div><div class="search"><button class="search-button" id="search-button"><p>Search</p><svg role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 19.9 19.7"><title>Search</title><g class="search-path" fill="none"><path stroke-linecap="square" d="M18.5 18.3l-5.4-5.4"></path><circle cx="8" cy="8" r="7"></circle></g></svg></button><div id="search-container"><div id="search-space"><input autocomplete="off" id="search-bar" name="search" type="text" aria-label="Search for something" placeholder="Search for something"/><div id="search-layout" data-preview="true"></div></div></div></div><button class="darkmode" id="darkmode"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" id="dayIcon" x="0px" y="0px" viewBox="0 0 35 35" style="enable-background:new 0 0 35 35" xml:space="preserve" aria-label="Dark mode"><title>Dark mode</title><path d="M6,17.5C6,16.672,5.328,16,4.5,16h-3C0.672,16,0,16.672,0,17.5 S0.672,19,1.5,19h3C5.328,19,6,18.328,6,17.5z M7.5,26c-0.414,0-0.789,0.168-1.061,0.439l-2,2C4.168,28.711,4,29.086,4,29.5 C4,30.328,4.671,31,5.5,31c0.414,0,0.789-0.168,1.06-0.44l2-2C8.832,28.289,9,27.914,9,27.5C9,26.672,8.329,26,7.5,26z M17.5,6 C18.329,6,19,5.328,19,4.5v-3C19,0.672,18.329,0,17.5,0S16,0.672,16,1.5v3C16,5.328,16.671,6,17.5,6z M27.5,9 c0.414,0,0.789-0.168,1.06-0.439l2-2C30.832,6.289,31,5.914,31,5.5C31,4.672,30.329,4,29.5,4c-0.414,0-0.789,0.168-1.061,0.44 l-2,2C26.168,6.711,26,7.086,26,7.5C26,8.328,26.671,9,27.5,9z M6.439,8.561C6.711,8.832,7.086,9,7.5,9C8.328,9,9,8.328,9,7.5 c0-0.414-0.168-0.789-0.439-1.061l-2-2C6.289,4.168,5.914,4,5.5,4C4.672,4,4,4.672,4,5.5c0,0.414,0.168,0.789,0.439,1.06 L6.439,8.561z M33.5,16h-3c-0.828,0-1.5,0.672-1.5,1.5s0.672,1.5,1.5,1.5h3c0.828,0,1.5-0.672,1.5-1.5S34.328,16,33.5,16z M28.561,26.439C28.289,26.168,27.914,26,27.5,26c-0.828,0-1.5,0.672-1.5,1.5c0,0.414,0.168,0.789,0.439,1.06l2,2 C28.711,30.832,29.086,31,29.5,31c0.828,0,1.5-0.672,1.5-1.5c0-0.414-0.168-0.789-0.439-1.061L28.561,26.439z M17.5,29 c-0.829,0-1.5,0.672-1.5,1.5v3c0,0.828,0.671,1.5,1.5,1.5s1.5-0.672,1.5-1.5v-3C19,29.672,18.329,29,17.5,29z M17.5,7 C11.71,7,7,11.71,7,17.5S11.71,28,17.5,28S28,23.29,28,17.5S23.29,7,17.5,7z M17.5,25c-4.136,0-7.5-3.364-7.5-7.5 c0-4.136,3.364-7.5,7.5-7.5c4.136,0,7.5,3.364,7.5,7.5C25,21.636,21.636,25,17.5,25z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" id="nightIcon" x="0px" y="0px" viewBox="0 0 100 100" style="enable-background:new 0 0 100 100" xml:space="preserve" aria-label="Light mode"><title>Light mode</title><path d="M96.76,66.458c-0.853-0.852-2.15-1.064-3.23-0.534c-6.063,2.991-12.858,4.571-19.655,4.571 C62.022,70.495,50.88,65.88,42.5,57.5C29.043,44.043,25.658,23.536,34.076,6.47c0.532-1.08,0.318-2.379-0.534-3.23 c-0.851-0.852-2.15-1.064-3.23-0.534c-4.918,2.427-9.375,5.619-13.246,9.491c-9.447,9.447-14.65,22.008-14.65,35.369 c0,13.36,5.203,25.921,14.65,35.368s22.008,14.65,35.368,14.65c13.361,0,25.921-5.203,35.369-14.65 c3.872-3.871,7.064-8.328,9.491-13.246C97.826,68.608,97.611,67.309,96.76,66.458z"></path></svg></button><div class="explorer"><button type="button" id="mobile-explorer" class="collapsed hide-until-loaded" data-behavior="collapse" data-collapsed="collapsed" data-savestate="true" data-tree="[{&quot;path&quot;:&quot;Definitions&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Functions&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Measure-Theory&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Measure-Theory/Sigma-Algebra&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Metric-Spaces&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Sets&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Statements&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Terminology&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Topological-Spaces&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Topological-Spaces/Induced&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Topological-Spaces/Terminologies&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Vector-Spaces&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw/Lecture-11&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw/Lecture-12&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw/Lecture-13&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Lectures&quot;,&quot;collapsed&quot;:true}]" data-mobile="true" aria-controls="explorer-content" aria-expanded="false"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-menu"><line x1="4" x2="20" y1="12" y2="12"></line><line x1="4" x2="20" y1="6" y2="6"></line><line x1="4" x2="20" y1="18" y2="18"></line></svg></button><button type="button" id="desktop-explorer" class="title-button" data-behavior="collapse" data-collapsed="collapsed" data-savestate="true" data-tree="[{&quot;path&quot;:&quot;Definitions&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Functions&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Measure-Theory&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Measure-Theory/Sigma-Algebra&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Metric-Spaces&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Sets&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Statements&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Terminology&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Topological-Spaces&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Topological-Spaces/Induced&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Topological-Spaces/Terminologies&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Definitions/Vector-Spaces&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw/Lecture-11&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw/Lecture-12&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Excalidraw/Lecture-13&quot;,&quot;collapsed&quot;:true},{&quot;path&quot;:&quot;Lectures&quot;,&quot;collapsed&quot;:true}]" data-mobile="false" aria-controls="explorer-content" aria-expanded="true"><h2>Explorer</h2><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="fold"><polyline points="6 9 12 15 18 9"></polyline></svg></button><div id="explorer-content"><ul class="overflow" id="explorer-ul"><li><div class="folder-outer open"><ul style="padding-left:0;" class="content" data-folderul><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"><button class="folder-button"><span class="folder-title">Definitions</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" 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 href="../Definitions/Functions/Metric" data-for="Definitions/Functions/Metric">Metric</a></li><li><a href="../Definitions/Functions/Power-Set" data-for="Definitions/Functions/Power-Set">Power Set</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/Measure-Theory"><button class="folder-button"><span class="folder-title">Measure Theory</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Definitions/Measure-Theory"><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/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 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/>
In other words ( ”):” ), for any open <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.9361em;"></span><span class="mord mover"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.9361em;"><span style="top:-3.3313em;"><span class="pstrut" style="height:3.3313em;"></span><span class="mord mover"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span class="svg-align" style="top:-3.7833em;"><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="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
5-6 9-10 13-.7 1-7.3 1-20 1H6z"></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="M200428 334
c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span></span></span></span></span><span style="top:-4.89em;"><span class="pstrut" style="height:3.3313em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight"></span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.10903em;">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: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:0.1514em;"><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:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"></span><span class="mord mathnormal">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:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>.</p>
<p>In fact, <a href="../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a> are unions of <a href="../Definitions/Metric-Spaces/Ball" class="internal alias" data-slug="Definitions/Metric-Spaces/Ball">balls</a>.</p>
<h1 id="topological-space">Topological Space<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#topological-space" 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> <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 set with a <a href="../Definitions/Topological-Spaces/Topology" class="internal" data-slug="Definitions/Topological-Spaces/Topology">Topology</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:0.1132em;">τ</span></span></span></span>.</p>
<blockquote class="callout example" data-callout="example">
<div class="callout-title">
<div class="callout-icon"></div>
<div class="callout-title-inner"><p>Trivial <a href="../Definitions/Topological-Spaces/Topology" class="internal" data-slug="Definitions/Topological-Spaces/Topology">Topology</a></p></div>
</div>
<div class="callout-content">
<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 the set. <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:0.1132em;">τ</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"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">}</span></span></span></span>.</p>
<p>Therefore <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>, then the only neighbour is <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><br/>
<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:0.2222em;"></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:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.</p>
</div>
</blockquote>
<blockquote class="callout example" data-callout="example">
<div class="callout-title">
<div class="callout-icon"></div>
<div class="callout-title-inner"><p>Discrete <a href="../Definitions/Topological-Spaces/Topology" class="internal" data-slug="Definitions/Topological-Spaces/Topology">Topology</a></p></div>
</div>
<div class="callout-content">
<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 the set. <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:0.1132em;">τ</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>.</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="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:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span><br/>
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">compact</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:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">finite</span></span></span></span></span></p>
<p>This topological space is always <a href="../Definitions/Topological-Spaces/Hausdorff" class="internal" data-slug="Definitions/Topological-Spaces/Hausdorff">Hausdorff</a> as it includes all the points 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> are included</p>
<blockquote class="callout note" data-callout="note">
<div class="callout-title">
<div class="callout-icon"></div>
<div class="callout-title-inner"><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">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:1em;vertical-align:-0.25em;"></span><span class="mopen">{</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 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:1em;vertical-align:-0.25em;"></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="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:0.6833em;"></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:0.8056em;vertical-align:-0.0556em;"></span><span class="mord"></span></span></span></span></p></div>
</div>
</blockquote>
<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">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:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if</span></span><span class="mspace"> </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 class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if</span></span><span class="mspace" style="margin-right:0.1667em;"></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 class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
</div>
</blockquote></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
c0.74,0,1.424-0.215,2.019-0.567l7.669,7.669C21.634,21.458,21,23.154,21,25s0.634,3.542,1.688,4.897L10.024,42.562
C8.958,41.595,7.549,41,6,41c-3.309,0-6,2.691-6,6s2.691,6,6,6s6-2.691,6-6c0-1.035-0.263-2.009-0.726-2.86l12.829-12.829
c1.106,0.86,2.44,1.436,3.898,1.619v10.16c-2.833,0.478-5,2.942-5,5.91c0,3.309,2.691,6,6,6s6-2.691,6-6c0-2.967-2.167-5.431-5-5.91
v-10.16c1.458-0.183,2.792-0.759,3.898-1.619l7.669,7.669C41.215,39.576,41,40.26,41,41c0,2.206,1.794,4,4,4s4-1.794,4-4
s-1.794-4-4-4c-0.74,0-1.424,0.215-2.019,0.567l-7.669-7.669C36.366,28.542,37,26.846,37,25s-0.634-3.542-1.688-4.897l9.665-9.665
C46.042,11.405,47.451,12,49,12c3.309,0,6-2.691,6-6S52.309,0,49,0z M11,9c0-1.103,0.897-2,2-2s2,0.897,2,2s-0.897,2-2,2
S11,10.103,11,9z M6,51c-2.206,0-4-1.794-4-4s1.794-4,4-4s4,1.794,4,4S8.206,51,6,51z M33,49c0,2.206-1.794,4-4,4s-4-1.794-4-4
s1.794-4,4-4S33,46.794,33,49z M29,31c-3.309,0-6-2.691-6-6s2.691-6,6-6s6,2.691,6,6S32.309,31,29,31z M47,41c0,1.103-0.897,2-2,2
s-2-0.897-2-2s0.897-2,2-2S47,39.897,47,41z M49,10c-2.206,0-4-1.794-4-4s1.794-4,4-4s4,1.794,4,4S51.206,10,49,10z"></path></svg></button></div><div id="global-graph-outer"><div id="global-graph-container" data-cfg="{&quot;drag&quot;:true,&quot;zoom&quot;:true,&quot;depth&quot;:-1,&quot;scale&quot;:0.9,&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;:true,&quot;enableRadial&quot;:true}"></div></div></div><div class="toc desktop-only"><button type="button" id="toc" class aria-controls="toc-content" aria-expanded="true"><h3>Table of Contents</h3><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="fold"><polyline points="6 9 12 15 18 9"></polyline></svg></button><div id="toc-content" class><ul class="overflow"><li class="depth-0"><a href="#open-sets" data-for="open-sets">Open Sets</a></li><li class="depth-1"><a href="#definition" data-for="definition">Definition</a></li><li class="depth-0"><a href="#topological-space" data-for="topological-space">Topological Space</a></li></ul></div></div><div class="backlinks"><h3>Backlinks</h3><ul class="overflow"><li><a href="../" class="internal">index</a></li></ul></div></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
socket.addEventListener('message', () => document.location.reload(true))
</script><script src="../postscript.js" type="module"></script></html>