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content/Math4201/Math4201_L20.md

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+# Math4201 Topology I (Lecture 20)
+
+## Quotient topology
+
+### More propositions
+
+#### Proposition for quotient maps in restrictions
+
+Let $X,Y$ be topological spaces and $p:X\to Y$ is surjective and open/closed. Let $A\subseteq X$ be saturated by $p$, ($p^{-1}(p(A))=A$).
+
+Then $q: A\to p(A)$ given by the restriction of $p$ is open/closed surjective map (In particular, it's a quotient map).
+
+<details>
+<summary>Proof</summary>
+
+$q$ is surjective and continuous. Now assume $p$ is open and we will show that $q$ is also open. Any open subspace of $A$ is given as $U\cap A$ where $U$ is open in $X$. By definition, $q(U\cap A)=p(U\cap A)=p(U)\cap p(A)$
+
+To see the second identity:
+
+1. $p(U\cap A)\subseteq p(U)\cap p(A)$
+
+$\forall y\in p(U\cap A)$, $y=p(x)$ with $x\in U\cap A$, since $x\in A$ and $x\in U$, $y=p(x)\in p(U)\cap p(A)$
+
+2. $p(U)\cap p(A)\subseteq p(U\cap A)$
+
+$\forall y\in p(U)\cap p(A)$, $y=p(x_1)$ with $x_1\in U$ and $y=p(x_2)$ with $x_2\in A$, since $x_1\in U$ and $x_2\in A$, $y=p(x_1)=p(x_2)\in p(U\cap A)$
+
+So $x_1=x_2\in U\cap A$, $y=p(x_1)=p(x_2)\in p(U)\cap p(A)$, $y\in p(U\cap A)$.
+
+Note that $p(U)\subseteq X$ is open by $p$ is an open map.
+
+So $p(U)\cap p(A)$ is open in $p(A)$.
+
+$q(U\cap A)=p(U\cap A)=p(U)\cap p(A)$ is open.
+
+So $q$ is open in $p(A)$.
+
+</details>
+
+### Simplicial complexes (extra chapter)
+
+#### Definition for simplicial complexes
+
+Simplicial complexes are topological space with simplices ($n$ dimensional triangles) as their building blocks.
+
+#### Definition for n dimensional simplex
+
+Let $v_0,\dots,v_n$ be points in $\mathbb{R}^m$ such that $v_n-v_0$, $v_{n-1}-v_0$, $\cdots$, and $v_1-v_0$ are linearly independent in $\mathbb{R}^m$. (in particular $n\leq m$).
+
+The $n$-dimensional simplex determined by $\{v_0,\dots,v_n\}$ is given as:
+
+$$
+\Delta^n\coloneqq [v_0,\dots,v_n]=\{t_0v_0+t_1v_1+\cdots+t_nv_n\vert t_i\geq 0, \sum_{i=0}^n t_i=1\}
+$$
+
+The coefficients $t_0,\dots,t_n$ are called barycentric coordinates.
+
+<details>
+<summary>Example of simplicial complex</summary>
+
+$n=0$,
+
+$\Delta^0=\{v_0\}$
+
+$n=1$,
+
+$\Delta^1=\{t_0v_0+t_1v_1\vert t_0+t_1=1\}$, this is the line segment between $v_0$ and $v_1$.
+
+$n=2$,
+
+$\Delta^2=\{t_0v_0+t_1v_1+t_2v_2\vert t_0+t_1+t_2=1\}$, this is the triangle with vertices $v_0,v_1,v_2$.
+
+</details>
+
+> [!NOTE]
+>
+> Every non-empty subset $\{v_{i_0},\dots,v_{i_k}\}$ of $\{v_0,\dots,v_n\}$ determines a $k$ dimensional simplex $[v_{i_0},\dots,v_{i_k}]\subseteq \Delta^n=[v_0,\dots,v_n]$. Inside the $n$ dimensional simplex $t_{i_0}v_{i_0}+\cdots+t_{i_n}v_{i_k}\in \Delta^n$. Where the coefficient $t_j$ of $v_j\notin \{v_{i_0},\dots,v_{i_n}\}$ is $0$.
+
+Any such $k$ dimensional simplex is called a face of the simplex $[v_{i_0},\dots,v_{i_n}]$.
+
+<details>
+<summary>Example of faces for simplicial complex</summary>\
+
+For a triangle $[v_0,v_1,v_2]$, the faces are $[v_0,v_1]$, $[v_0,v_2]$, and $[v_1,v_2]$ (the edges of the triangle).
+
+</details>
+
+#### Definition for abstract simplicial complex
+
+Let $V$ be a finite or countable set, an abstract simplicial complex on $V$ is a collection of **finite non-empty subset** of $V$, denoted by $K$. And the two conditions are satisfied:
+
+1. If $\sigma\in K$ and $\tau\subseteq \sigma$, then $\tau\in K$.
+
+2. For any $v\in V$, $\{v\}\in K$.
+
+<details>
+<summary>Example of abstract simplicial complex</summary>
+
+Let $V=\{a,b,c,d\}$.
+
+If we want to include $\{a,b,c\}$, then we need to include $\{a,b\}$ and $\{b,c\}$, so we have $K=\{\{a,b,c\},\{a,b\},\{b,c\},\{a\},\{b\},\{c\},\{d\}\}$ is an abstract simplicial complex. 
+
+</details>
+
+#### Topological realization of abstract simplicial complex
+
+Let $\bigsqcup_{\sigma\in K}\Delta^{|\sigma|-1}$ be the disjoint union of all $|\sigma|-1$ dimensional simplices in $K$.
+
+$$
+\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta^{|\sigma|-1}
+$$
+
+We use subspace topology to define a topology on $\Delta^n$ and the union of such topology for each $\Delta^{|\sigma|-1}$ defines a topology on $\tilde{X_k}$.
+
+We define the equivalence relation $x\in \Delta_{\sigma}^{|\sigma|-1}\sim x'\in \Delta_{\sigma'}^{|\sigma'|-1}$ if $x\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_{\sigma}^{|\sigma|-1}$. and $x'\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_{\sigma'}^{|\sigma'|-1}$.
+
+are the sample points of $\Delta_{\sigma\cap \sigma'}^{|\sigma\cap \sigma'|-1}$.
+
+$X_K$ is the quotient space of $\tilde{X_k}$ by the equivalence relation.
+
+Continue next time.

content/Math4201/_meta.js

@@ -23,4 +23,5 @@ export default {
     Math4201_L17: "Topology I (Lecture 17)",
     Math4201_L18: "Topology I (Lecture 18)",
     Math4201_L19: "Topology I (Lecture 19)",
+    Math4201_L20: "Topology I (Lecture 20)",
 }