From d8b699ec3c91bb1d45ca4a7da41db534670d4787 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Fri, 16 Jan 2026 11:52:01 -0600
Subject: [PATCH] updates

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 content/Math4202/Math4202_L3.md | 82 +++++++++++++++++++++++++++++++++
 content/Math4202/_meta.js       |  1 +
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 create mode 100644 content/Math4202/Math4202_L3.md

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+# Math4202 Topology II (Lecture 3)
+
+## Reviewing quotient map
+
+### Quotient map from equivalence relation
+
+Consider $X,Y$ be two topological space and $A\subset X$, where $f:A\to Y$ is a function.
+
+Then the disjoint union $X\sqcup Y /_{a\sim f(a)}$ is a quotient space of $X\sqcup Y$ by the equivalence relation $a\sim f(a)$
+
+Consider $e^n$ be the n dimensional closed ball (n-cells)
+
+$$
+e^n=\{x\in \mathbb{R}^n:\sum_{i=1}^n x_i^2\leq 1\}
+$$
+
+and $\partial e^n=A$ be the $n-1$ dimensional sphere.
+
+#### CW complex
+
+Let $X_0$ be arbitrary set of points.
+
+Then we can create $X_1$ by
+
+$$
+X_1=\{(e_\alpha^1,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^1\to X_0\}
+$$
+
+where $\varphi$ is a continuous map, and $e_\alpha^1$ is a $1$-cell (interval).
+
+$$
+X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
+$$
+
+and $e_\alpha^2$ is a $2$-cell (disk). (mapping boundary of disk to arc (like a croissant shape, if you try to preserve the area))
+
+The higher dimensional folding cannot be visualized in 3D space.
+
+$$
+X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1}
+$$
+
+<details>
+<summary>Example of CW complex construction</summary>
+
+$X_0=a$
+
+$X_1=$ circle, with end point and start point at $a$
+
+$X_2=$ sphere (shell only), with boundary shrinking at the circle create by $X_1$
+
+---
+
+$X_0=a$
+
+$X_1=a$
+
+$X_2=$ ballon shape with boundary of circle collapsing at $a$
+</details>
+
+#### Theorem of quotient space
+
+Let $p:X\to Y$ be a quotient map, let $Z$ be a space and $g:X\to Z$ be a map that is constant on each set $p^{-1}(y)$ for each $y\in Y$.
+
+Then $g$ induces a map $f: X\to Z$ such that $f\circ p=g$.
+
+The map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient map if and only if $g$ is a quotient map.
+
+## Imbedding of Manifolds
+
+### Manifold
+
+#### Definition of Manifold
+
+An $m$-dimensional **manifold** is a topological space  $X$ that is
+
+1. Hausdorff
+2. With a countable basis such that each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
+
+> [!NOTE]
+>
+> Try to find some example that satisfies some of the properties above but not a manifold.
diff --git a/content/Math4202/_meta.js b/content/Math4202/_meta.js
index 4162f5b..13ed57f 100644
--- a/content/Math4202/_meta.js
+++ b/content/Math4202/_meta.js
@@ -5,4 +5,5 @@ export default {
     },
     Math4202_L1: "Topology II (Lecture 1)",
     Math4202_L2: "Topology II (Lecture 2)",
+    Math4202_L3: "Topology II (Lecture 3)",
 }