Quartz sync: Mar 1, 2025, 2:26 PM

content/Definitions/Cauchy Sequence.md

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+# 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.
+
+Furthermore, here, using something like $\lim_{ n \to \infty } (-1)^n$ is nearly impossible to know what the value would be as $\infty$ is neither even or odd.

content/Definitions/Cauchy-Schwarz Inequality.md

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+# Definition
+$$| (u|v) | \leq \| \, u \, \|_{2} \times \| \, v \, \|_{2}$$
+$$\| \, u + v \, \|_{2} \leq \| \, u \, \|_{2} + \| \, v \, \|_{2}$$
+> [!example] Proof of Cauchy-Schwartz
+> Insert $a \equiv - \frac{\overline{(u | v)}}{\| \, u \, \|_{2}^2}$ for $u \neq 0$ into
+> $$f(a) \equiv |a|^2 \times \| \, u \, \|_{2}^2 + \text{Re}(a \times (u | v)) + \| \, v \, \|_{2}^2 = \| \, au + v \, \|_{2}^2 \geq 0$$
+> [[QED]]
+

content/Definitions/Functions/Characteristic Function.md

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+$\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}$, so $\varkappa_{y} : X \to \{ 0, \, 1 \}$ defined as:
+$$\varkappa_{y}(x) = \begin{cases}
+1, & \text{if}\ x \in Y\\ 0, & \text{if}\ x \notin Y
+\end{cases}$$

content/Definitions/Functions/Direct Product.md

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+$$X \times Y = \{ x: \{ 1, 2 \} \to x_{1} \cup x_{2} | x_{1} X_{1} \wedge x_{2} \in X_{2} \}$$
+where $X = X_{1}$ and $Y = X_{2}$, basically $(X_{1}, X_{2})$.

content/Definitions/Functions/Inverse Function.md

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+$$f^{-1} : Y \to X$$
+such that
+$$ff^{-1} = I_{x} \; \land f^{-1}f = I_{y}$$
+$$\exists f^{-1} \iff f \; \text{bijective}$$

content/Definitions/Functions/Metric.md

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+# Definition
+A function which measures distances between two points in a [[Metric Space]].

content/Definitions/Functions/Power Set.md

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+# 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.
+# Example
+$$X = \{ 1, \, 2 \}$$
+$$\wp(X) = \{ \emptyset, \, \{ 1 \}, \, \{ 2 \}, \, X \}$$

content/Definitions/Hilbert Spaces.md

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+# Definition
+These are [[Banach Space|Banach Spaces]] with norms given by an [[Inner Product]].
+
+> [!note] The norm is defined as:
+> $$\| \, v \, \|_{2} \equiv \sqrt{ (v | v) }$$
+> $$d(u, v) = \| \, u - v \, \|_{2}$$
+

content/Definitions/Inner Product.md

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+# 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 $\overline{(u | v)} = (v | u)$
+> >[!example]-
+> > $$(w | au + bv) = \overline{(au + bv | w)} = \overline{a(u | w) + b (v | w)} = \bar{a} \overline{(u | w)} + \bar{b} \overline{( v | w)} = \dots$$
+>
+> and $(v | v) \geq 0$ and $(u | u) = 0 \implies u = 0$

content/Definitions/Least Upper Bound Property.md

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+$$X \subset \mathbb{R}$$
+$$\exists c \in \mathbb{R} \; \text{such that}$$
+$$x \lt c, \forall x \in X$$
+
+
+$$sup(\langle 0, 1 \rangle) = 1 \notin \langle 0, 1 \rangle$$
+$$sup(\langle 0, 1 ] \,)$$
+
+
+$$sup(\mathbb{R}) = \infty$$

content/Definitions/Linear Map.md

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+# Definition
+$$L: V \to \mathbb{C}^n \; \text{bijective}$$
+$$L (ax + by) = a \times L(x) + b \times L(y), \; \forall x, \, y \in V, \, a; \, b \in \mathbb{C}$$
+$$\| \, L(x) \, \|  = \| \, x \, \|, \; (L(x) | L(y)) = (x | y)$$

content/Definitions/Measure Theory/Sigma-Algebra/Borel Measurable.md

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+# Definition
+Say $f : X \to Y$ is [[Continuous|continuous]], $\implies f$ is Borel measurable.

content/Definitions/Measure Theory/Sigma-Algebra/Borel Sets.md

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+# Definition
+The $\sigma$-algebra generated by the [[Open Sets|open sets]] in a [[Topological Space|topological space]] $X$ is the $\sigma$-algebra of **Borel Sets** of $X$.

content/Definitions/Measure Theory/Sigma-Algebra/Measurable.md

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+# Definition
+$f: X \to Y$ ([[Topological Space]]) is **measurable** if $f^{-1}(V) \in M$ $\forall \text{ open } V \subset Y$.

content/Definitions/Measure Theory/Sigma-Algebra/Measure.md

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+# Definition
+A **measure** on $X$ is a function $\mu : M \to [0,\infty]$ such that:
+1. $\mu(\emptyset) = 0$
+2. $\mu(\cup^{\infty}_{n=1}A_{n}) = \Sigma^{\infty}_{n=1} \mu(A_{n})$ (for pairwise disjoint $A_{n} \in M$)
+Then we say $X$ is a measure space.

content/Definitions/Measure Theory/Sigma-Algebra/Sigma-Algebra.md

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+# Definition
+A $\sigma$-algebra in a set $X$ is a collection of subsets, so called measurable sets, of $X$ such that (the requirements are):
+1. $X \in M$
+2. $A \in M \implies A^{\complement} \in M$ ($X^{\complement} = \emptyset \in M$)
+3. $A_{n} \in M \implies \cup^{\infty}_{n=1} A_{n} \in M$ ($\implies \cap^{\infty}_{n=1} A_{n} = (\cup^{\infty}_{n=1}A^{\complement})^{\complement} \in M)$)
+# Related Terminologies/Functions
+- [[Measurable]]
+- [[Measure]]

content/Definitions/Metric Spaces/Ball.md

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+# Definition
+Let $d$ be a [[Metric|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 $\underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N$.

content/Definitions/Metric Spaces/Interior Point.md

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+# Definition
+Points such that small enough balls centred around them are contained in $A$.

content/Definitions/Metric Spaces/Metric Space.md

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+# Definition
+A [[Metric]] on a set $X$ is a function $d : X \times X \to [ \, 0, \infty \rangle$ such that
+1. $d(x,y) = d(y, x), \; \forall x,y \in X$
+2. $d(x, y) = 0 \iff x = y$
+3. $d(x,z) \leq d(x,y) + d(y, z)$ (think of this as a triangle and Pythagoras' Theorem)
+Think of $d(x, y)$ as the distance between $x$ and $y$.

content/Definitions/Nets.md

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+$$\{ x_{i} \}_{i \in I} \; \text{NET} \; \underbrace{I}_{\text{VFOs}} \to X$$
+$$x \in \overline{X} \iff \exists \; \text{NET} \; \underbrace{x_{i}}_{\in X} \to x$$
+$I$ = neighbourhoods of $x$ with $A \geq B \iff A \subset B$.
+
+$\{ \emptyset, X \}$ - all nets in $X$ will converge to all points in $X$.

content/Definitions/Norm.md

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+# Definition
+A **norm** on $V$ is a map $\|\cdot\| : V \to [ \, 0, \infty \rangle$ such that
+1. $\| u + v \| \leq \| u \| + \| v \|, \; \forall u,v \in V$
+2. $\| c \times v \| = | c | \times \| v \|, \; \forall v \in V, \, \forall c \in \mathbb{C}$
+3. $\| v \| = 0 \implies v = 0$
+
+Think of $\| v \|$ as the length of $v$.
+## Norm of 0
+$$\| 0 \| = \| 0 \times u \| = \| c \times u \| = \| 0 \| \times \|u \| = 0 \times \| u \| = 0$$

content/Definitions/Number Field.md

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+$$a \neq 0, \; \frac{1}{a}$$
+$$ab = ba$$
+$$a(b+c) = ab + ac$$
+$$a \gt b \iff a+1 \gt b + 1$$

content/Definitions/Period of a Fraction.md

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+# Definition
+The period of a fraction is the digits that repeat themselves.
+
+The digits can be from $0-9$, and the length of the period is determined by the denominator, n in $\frac{a}{n}$ .
+# Examples
+$\frac{22}{7}= 3.142857142857\dots$ where the period here is $142857$ and has a length 6
+$\frac{1}{2} = 0.5000000\dots$ has the period $0$, with the length being 1.

content/Definitions/Rational Cauchy Sequences.md

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+# Definition
+They are sequences $\{ x_{n} \subset \mathbb{Q} \}$ such that
+$$\forall k \in \mathbb{N} \; \exists N \in \mathbb{N}$$
+Such that
+$$|x_{m} - x_{n}| \lt \frac{1}{k}, \; \forall m, n \gt N.$$

content/Definitions/Sets/Complex Numbers.md

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+$$z = a + ib$$
+$$z = r(\cos \theta + i\sin \theta)$$
+$$z = re^{i \theta}$$
+# Operations
+## Multiplying Vectors
+$$z_{1} \times z_{2} = (r_{1} \times r_{2})(\cos(\theta_{1} + \theta 2) + i \sin (\theta_{1} + \theta_{2}))$$

content/Definitions/Sets/Open Cover.md

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+# Definition
+[[Open Sets]] with union $X$

content/Definitions/Sets/Open Map.md

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+# Definition
+An **open map** takes one [[Open Sets|open set]] and maps it to another [[Open Sets|open set]].

content/Definitions/Sets/Open Sets.md

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+# Definition
+An open set $A \subset X$ in $(X, d)$ (means set with a metric) consists only of [[Interior Point|interior points]].
+
+Then a sequence converges to $x \in X$ $\iff$ it eventually belongs to any [[Open Sets|open set]] containing $x$.

content/Definitions/Statements/And.md

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+$P \land Q$
+# Truth Table
+
+| P   | Q   | =   |
+| --- | --- | --- |
+| F   | F   | F   |
+| F   | T   | F   |
+| T   | F   | F   |
+| T   | T   | T   |

content/Definitions/Statements/Implies.md

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+$P \implies Q$
+# Truth Table
+
+| P   | Q   | =   |
+| --- | --- | --- |
+| F   | F   | T   |
+| F   | T   | T   |
+| T   | F   | F   |
+| T   | T   | T   |

content/Definitions/Statements/Not.md

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+$¬P$
+The symbol ($¬$) is called negation
+# Truth Table
+
+| P   | =   |
+| --- | --- |
+| F   | T   |
+| T   | F   |

content/Definitions/Statements/Or.md

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+$P \lor Q$
+# Truth Table
+
+| P   | Q   | =   |
+| --- | --- | --- |
+| F   | F   | F   |
+| F   | T   | T   |
+| T   | F   | T   |
+| T   | T   | T   |

content/Definitions/Subcover.md

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+# Definition
+Any subcollection with union $X$

content/Definitions/Subnet.md

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+# Definition
+A **subnet** of a [[Nets|net]] $f: I \to X$ is a [[Nets|net]] $g: J \to X$ and a map $h : J \to I$ such that $g = f \circ h$ and such that $\forall i \in I$ $\exists j \in J$ with $h(j') \geq i \; \forall j' \geq j$.
+# Example
+> [!example]
+> ![[Drawing 2025-02-13 11.47.33.excalidraw]]
+> **Sequence:**
+> $$f : I = \mathbb{N} \to \mathbb{R} : f(n) = x_{n}$$
+> **Subsequence:**
+> $$\{ \underbrace{x_{1}}_{= x_{1}}, \, \underbrace{x_{3}}_{= x_{2}}, \, \underbrace{x_{5}}_{= x_{3}}, \, \underbrace{x_{7}}_{= x_{4}}, \, \dots \} \to 1$$
+> **Definition:**
+> $g : J = \mathbb{N} \to \mathbb{R}$ by $g(j) = x'_{j} = x_{2j-1} = f(2j-1) = f \circ h(j)$, where $h : J \to I$ is defined by $h(j) = 2j-1$.

content/Definitions/Terminology/Algebraically Complete.md

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+# For Complex Numbers
+Any equation
+$$a_{n} z^n + \dots + a_{1} z^1 + a_{0} = 0$$
+with $a_{i} \in \mathbb{C}$ has $n$ roots, or solutions, counted with multiplicity.

content/Definitions/Terminology/Bijective.md

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+# 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 $\text{House}$.
+
+Another example could be
+$$\{ 1, \dots, 5 \} \xrightarrow{f^{\text{\; Bijective}}} X$$

content/Definitions/Terminology/Bounded.md

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+# Definition
+$\exists M$ such that $\| \, x \, \| M, \; \forall x \in A$.
+![[Pasted image 20250130113939.png]]

content/Definitions/Terminology/Countable.md

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+# What is countable?
+## Natural Numbers
+Quite simple
+$$\mathbb{N} = \{ 1, 2, 3, 4, \dots \}$$
+Which you can just add it up every time
+## Integer Numbers
+$Z$ can be countable as you can go
+$$1 \to 0$$
+$$2 \to 1$$
+$$3 \to -1$$
+$$4 \to 2$$
+$$5 \to -2$$
+and so on...
+
+Here they start at 0, then go 1, -1, 2, -2, etc...
+## Rational Numbers
+They are countable, but it is a lot more work to show

content/Definitions/Terminology/Injective.md

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+$$f: \mathbb{R} \to \mathbb{R}$$
+Usually a function would be mapped like:
+$$G(f) = \{ (x, f(x)) | x \in X \}$$
+# Definition
+A horizontal line going through a function (on a graph) should only intersect the function only once

content/Definitions/Terminology/QED.md

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+Can be written as "Q.E.D." or "QED".
+
+It is shortened in Latin from "quod erat demonstrandum" (that which was to be demonstrated).
+
+# Definition
+A notation which is often placed at the end of a mathematical proof to indicate its completion

content/Definitions/Terminology/Surjective.md

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+$$y \in Y$$
+$$\exists x \in X$$
+$$\text{such that} \; f(x) = y$$

content/Definitions/Topological Spaces/Continuous.md

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+# 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 Sets|open]] if $f(B)$ is [[Open Sets|open]] and $\forall$ [[Open Sets|open]] $B \subset X$.
+
+If $f$ is a [[Bijective|bijection]] that is both **continuous** and [[Open Sets|open]], it is a [[Homeomorphic|homeomorphism]], and $X$ and $Y$ are [[Homeomorphic|homeomorphic]], written $X \simeq Y$; they are the 'same' as [[Topological Space|topological spaces]].
+## In-depth Definition
+A function $f : X \to Y$ between [[Topological Space|topological spaces]] is **continuous at $x \in X$** if for every  neighbourhood $A$ of $f(x)$, we can find a neighbourhood $B$ of $x$ such that $f(B) \subset A$, or $B \subset f^{-1}(A)$.

content/Definitions/Topological Spaces/Hausdorff.md

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+# Definition
+If there are two points $x$ and $y$ in a [[Topological Space|topological space]] $X$ that can be separated by neighbourhoods if there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $y$, such that $U$ and $V$ are disjoint $U \cup V = \emptyset$.
+
+$X$ is a **Hausdorff space** if any two distinct points in $X$ are separated by neighbourhoods.

content/Definitions/Topological Spaces/Induced/Initial Topology.md

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+# Definition
+The **initial topology** on $X$ induced by a family of functions $f : X \to Y_{f}$ into [[Topological Space|topological spaces]] $Y_{f}$ is the [[Weakest Topology|weakest topology]]on $X$ making all these functions [[Continuous|continuous]].
+
+Here: $F = \{ f^{-1} \; | \; f : X \to Y_{f}, \; A \; \text{open in} \; Y_{f} \}$.

content/Definitions/Topological Spaces/Induced/Product Topology.md

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+# Definition
+The **product topology** on $\Pi X_{\lambda}$, $X_{\lambda}$ [[Topological Space|topological spaces]], is the [[Initial Topology|initial topology]] induced by the family of projections $\Pi_{\lambda}$
+> [!note] What is $\Pi_{\lambda}$?
+> $\pi_{\lambda} : \Pi_{\lambda' \in \wedge} X_{\lambda'} \to X_{\lambda}$
+> $\pi_{\lambda}(f) = f(\lambda)$
+> $\pi_{\lambda}((X_{\lambda'})) = x_{\lambda}$
+
+$$\underbrace{\Pi_{\lambda \in \wedge} X_{\lambda} \equiv}_{\in (x_{\lambda})_{\lambda \in \wedge}} \{ f : \wedge \to \cup_{\lambda \in \wedge} X_{\lambda} \; | \; \underbrace{f(\lambda)}_{x_{\lambda}} \in X_{\lambda} \}$$
+# Example
+Product of 2 [[Topological Space|topological spaces]]: $x_{1} \times x_{2}$
+$= \{ (x_{1}, x_{2}) \; | \; x_{i} \in X_{i} \}$
+$\pi_{1}((x_{1}, x_{2})) = x_{1}$
+![[Drawing 2025-02-24 12.47.32.excalidraw]]

content/Definitions/Topological Spaces/Induced/Separating Points.md

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+# Definition
+A family $F$ of functions on a set **separating points** $x \neq y$ in the set if $f(x) \neq f(y)$ for some $f \in F$

content/Definitions/Topological Spaces/Induced/Weakest Topology.md

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+# Definition
+Given $F \subset \wp(X)$. The **weakest topology** on $X$ that contains $F$ is the intersection of all the [[Topology|topologies]] that contains $F$. This is a [[Topology|topology]], and consists of $\emptyset$, $X$, and all unions of finite intersections of members from $F$.
+
+> [!example]
+> $F \subset \tau$
+> $\textvisiblespace \cap \tau \ni U_{i} \implies U_{i} \in \tau$
+> $\implies \cap_{i \in F} \; U_{i} \in \tau \implies \cap U_{i} \in \cap_{F \in \tau} \;\tau$
+> 
+> $x_{i} \to x$
+> $\exists j$ such that $x_{i} = x, \; i \geq j$.
+

content/Definitions/Topological Spaces/Terminologies/Compact.md

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+# Definition
+$X$ is **compact** if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]].

content/Definitions/Topological Spaces/Terminologies/Connected Component.md

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+# Definition
+A **connected component** of a [[Topological Space|topological space]] is the union of all [[Connected|connected]] subsets that contain a given point. It itself is [[Connected|connected]].

content/Definitions/Topological Spaces/Terminologies/Connected.md

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+# Definition
+A [[Topological Space|topological space]] is **connected** if it is not a union of two non-empty [[Open Sets|open sets]].
+
+i.e. if you draw the two non-empty [[Open Sets|open sets]] on the graph, if you have to lift your pen, it will not be connected.
+
+# Examples
+Not connected:
+$$\langle 0, 1 ]  \cup [ 2, 5 \rangle$$
+Connected:
+$$\langle 0, 1 ] \cup [ 0.5, 5 \rangle = \langle 0, 5 \rangle$$

content/Definitions/Topological Spaces/Topological Space.md

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+# Definition
+Let $X$ be a topological space
+1. A **neighbourhood** of $x \in X$ is an [[Open Sets|open set]] $A$ with $x \in A$.
+2. $X$ is Hausdorff if for $x \neq y\; \exists \, \text{neighbourhoods} \, A, B$ of $x$ and $y$ respectively, such that $A \cap B = \emptyset$.
+3. $A \subset X$ is **closed** if $A^{\complement}$ is open.
+4. The **closure** (denoted by a bar over the set) $\overline{Y}$ of a subset $Y \subset X$ is the intersection of all closed subsets of $X$ that contain $Y$.
+5. $X$ is **[[Compact|compact]]** if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]].
+6. $X$ is **locally compact** if any $x \in X$ has a neighbourhood with compact closure.
+7. $X$ is $\sigma$-compact if it is a countable union of compact subsets with respect to the **relative topology**, i.e. an [[Open Sets|open set]] of a subset $Z$ of $X$ is of the type $Z \cap A$ where $A$ is open in $X$.
+## Dense
+Say $Y$ is **dense** in $X$ if $\overline{Y} = X$. If $Y$ is countable.
+## Separable
+Say $\overline{Y} = X$, then we say that $X$ is **separable**.

content/Definitions/Topological Spaces/Topology.md

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+# Definition
+A collection of subsets of $X$, called [[Open Sets|open sets]], such that:
+1. $X, \, \emptyset \in \tau$
+2. Any union of sets from $\tau$ will be in $\tau$.
+> [!info]-
+> $$y, z \in \tau \implies y \cup z \in \tau$$
+> $$x_{i} \in \tau \implies \cup_{i \in I} X_{i} \in \tau$$
+3. Any **finite** intersection of sets from $\tau$ will be in $\tau$.
+> [!info]-
+> $$y \cap z \in \tau$$
+> $$\cap_{i \in I} X_{i} \notin \tau$$
+
+# Examples
+> [!example]
+> The **topology induced by a metric** on $X$ is the collection of all unions of [[Ball|balls]].
+
+> [!example] Reasoning for having point/rule 3 in [[#Definition]]
+> Consider the topology on $\mathbb{R}$ induced by the usual distance.
+> $$B_{r}(x) = \langle x - r, x + r \rangle$$
+> Note:
+> $$\cap_{n \in \mathbb{N}} \langle -\frac{1}{n}, \frac{1}{n} \rangle = \{ 0\}$$
+> ($\cap_{n \in \mathbb{N}}$ is an infinite intersection of all numbers (in $\mathbb{N}$))
+> But the reason why this is not $= \{ 0, \varepsilon \}$ is a finite amount of intersections
+> $\varepsilon \notin \langle -\frac{1}{n}, \frac{1}{n} \rangle$ for $n \gt \frac{1}{\varepsilon}$
+

content/Definitions/Topological Spaces/Tychonoff Theorem.md

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+Can be abbreviated as "THM".
+# Definition
+The product of compact spaces is compact in the [[Product Topology|product topology]]

content/Definitions/Vector Spaces/Complex Vector Space.md

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+# Definition
+A **complex vector space** is a set $V$ with addition $u + v$ of vectors $u$, $v$, and scalar multiplication $a \times v, \; a \in \mathbb{C}, \; v \in V$, satisfying the [[Properties of a Vector Space]]

content/Definitions/Vector Spaces/Linear Basis.md

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+# Definition
+A **linear basis** of $V$ is a subset $\{ v_{i} \} \subset V$ such that every vector can be written uniquely as $\Sigma_{i} c_{i} v_{i}$ for finitely many non-zero $c_{i} \in \mathbb{C}$.

content/Definitions/Vector Spaces/Normed Vector Space.md

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+# Definition
+Have a metric given by the [[Norm]] on a vector space $V$ as $d(u, v) \equiv \|u - v\| \in [ \, 0, \infty \rangle$.

content/Definitions/Vector Spaces/Properties of a Vector Space.md

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+- $u + v = v + u$,
+- $(u+v) + w = u + (v + w)$,
+- $u + 0 = u$,
+- $u + (-u) = 0$,
+- $a (u + v) = a \times u + a \times v$,
+- $(a + b) \times v = av + bv$,
+- $a(bv) = (ab) \times v$,
+- $1 \times v = v$.
+
+These can be used for Complex, Real, or Rational vector spaces.