From 3483618c073645304791bf8907eaa68b9a211369 Mon Sep 17 00:00:00 2001
From: Trance-0 <60459821+Trance-0@users.noreply.github.com>
Date: Fri, 3 Oct 2025 11:52:56 -0500
Subject: [PATCH] Update Math4201_L17.md

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 # Math4201 Topology I (Lecture 17)
 
+## Quotient topology
+
+How can we define topologies on the space obtained points in a topological space?
+
+### Quotient map
+
+Let $(X,\mathcal{T})$ be a topological space. $X^*$ is a set and $q:X\to X^*$ is a surjective map.
+
+The quotient topology on $X^*$ is defined as follows:
+
+$$
+\mathcal{T}^* = \{U\subseteq X^*\mid q^{-1}(U)\in \mathcal{T}\}
+$$
+
+$U\subseteq X^*$ is open if and only if $q^{-1}(U)$ is open in $X$.
+
+In particular, $q$ is continuous map.
+
+#### Definition of quotient map
+
+$q:X\to X^*$ defined above is called a **quotient map**.
+
+#### Definition of quotient space
+
+$(X^*,\mathcal{T}^*)$ is called the **quotient space** of $X$ by $q$. 
+
+### Typical way of constructing a surjective map
+
+#### Equivalence relation
+
+$\sim$ is a subset of $X\times X$ satisfying:
+
+- reflexive: $\forall x\in X, x\sim x$
+- symmetric: $\forall x,y\in X, x\sim y\implies y\sim x$
+- transitive: $\forall x,y,z\in X, x\sim y\text{ and } y\sim z\implies x\sim z$
+
+#### Equivalence classes
+
+Check equivalence relation.
+
+For $x\in X$, the equivalence class of $x$ is denoted as $[x]\coloneqq \{y\in X\mid y\sim x\}$.
+
+$X^*$ is the set of all equivalence classes on $X$.
+
+$q:X\to X^*$ is defined as $q(x)=[x]$ will be a surjective map.
+
+<details>
+<summary>Example of surjective maps and their quotient spaces</summary>
+
+Let $X=\mathbb{R}^2$ and $(s,t)\sim (s',t')$ if and only if $s-s'$ and $t-t'$ are both integers.
+
+This space as a topological space is homeomorphic to the torus.
+
+---
+
+Let $X=\{(s,t)\in \mathbb{R}^2\mid s^2+t^2\leq 1\}$ and $(s,t)\sim (s',t')$ if and only if $s^2+t^2$ and $s'^2+t'^2$. with subspace topology as a subspace of $\mathbb{R}^2$.
+
+This space as a topological space is homeomorphic to the spherical shell $S^2$.
+
+</details>
+
+We will show that the quotient topology is a topology on $X^*$.
+
+<details>
+<summary>Proof</summary>
+
+We need to show that the quotient topology is a topology on $X^*$.
+
+1. $\emptyset, X^*$ are open in $X^*$.
+
+$\emptyset, X^*$ are open in $X^*$ because $q^{-1}(\emptyset)=q^{-1}(X^*)=\emptyset$ and $q^{-1}(X^*)=X$ are open in $X$.
+
+2. $\mathcal{T}^*$ is closed with respect to arbitrary unions.
+
+$$
+q^{-1}(\bigcup_{\alpha \in I} U_\alpha)=\bigcup_{\alpha \in I} q^{-1}(U_\alpha)
+$$
+
+3. $\mathcal{T}^*$ is closed with respect to finite intersections.
+
+$$
+q^{-1}(\bigcap_{\alpha \in I} U_\alpha)=\bigcap_{\alpha \in I} q^{-1}(U_\alpha)
+$$
+
+</details>
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