From 90b2f582cc3d90c8ebb3d61613bda3c64af365f7 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Fri, 23 Jan 2026 12:39:53 -0600 Subject: [PATCH] updates --- content/Math4202/Math4202_L5.md | 154 ++++++++++++++++++++++++++++++++ content/Math4202/_meta.js | 1 + 2 files changed, 155 insertions(+) create mode 100644 content/Math4202/Math4202_L5.md diff --git a/content/Math4202/Math4202_L5.md b/content/Math4202/Math4202_L5.md new file mode 100644 index 0000000..da4516e --- /dev/null +++ b/content/Math4202/Math4202_L5.md @@ -0,0 +1,154 @@ +# Math4202 Topology II (Lecture 5) + +## Manifolds + +### Imbedding of Manifolds + +> [!NOTE] +> +> Suppose $f: X \to Y$ is an injective continuous map, where $X$ and $Y$ are topological spaces. Let $Z$ be the image set $f(X)$, considered as a subspace of $Y$, then the function $f’: X \to Z$ obtained by restricting the range of f is bijective. If f happens to be a homeomorphism of X with Z, we say that the map $f: X \to Y$ is a topological imbedding, or simply imbedding, of X in Y. + +Recall from last lecture + +#### Whitney's Embedding Theorem + +If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$ for some positive integer $N$. + +_In general, $X$ is not required to be compact. And $N$ is not too big. For non compact $X$, $N\leq 2m+1$ and for compact $X$, $N\leq 2m$._ + +#### Definition for partition of unity + +Let $\{U_i\}_{i=1}^n$ be a finite open cover of topological space $X$. An indexed family of **continuous** function $\phi_i:X\to[0,1]$ for $i=1,...,n$ is said to be a **partition of unity** dominated by $\{U_i\}_{i=1}^n$ if + +1. $\operatorname{supp}(\phi_i)=\overline{\{x\in X: \phi_i(x)\neq 0\}}\subseteq U_i$ (the closure of points where $\phi_i(x)\neq 0$ is in $U_i$) for all $i=1,...,n$ +2. $\sum_{i=1}^n \phi_i(x)=1$ for all $x\in X$ (partition of function to $1$) + +#### Existence of finite partition of unity + +Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair of closed sets in $X$ can be separated by two open sets in $X$). + +Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$. + +_A more generalized version, If the space is paracompact, then there exists a partition of unity dominated by $\{U_i\}_{i\in I}$ with locally finite. (Theorem 41.7)_ + +
+Proof for Whithney's Embedding Theorem + +Since $X$ is a $m$ compact manifold, $\forall x\in X$, there is an open neighborhood $U_x$ of $x$ such that $U_x$ is homeomorphic to $\mathbb{R}^m$. That means there exists $\varphi_i:U_x\to \varphi(U_x)\subseteq \mathbb{R}^m$. + +Where $\{U_x\}_{x\in X}$ is an open cover of $X$. Since $X$ is compact, there is a finite subcover $\bigcup_{i=1}^k U_{x_i}=X$. + +Apply the [existence of partition of unity](#existence-of-finite-partition-of-unity), we can find a partition of unity dominated by $\{U_{x_i}\}_{i=1}^k$. With family of functions $\phi_i:\mathbb{R}^d\to[0,1]$. + +Define $h_i:X\to \mathbb{R}^m$ by + +$$ +h_i(x)=\begin{cases} +\phi_i(x)\varphi_i(x) & \text{if }x=x_i\\ +0 & \text{otherwise} +\end{cases} +$$ + +We claim that $h_i$ is continuous using pasting lemma. + +On $U_i$, $h_i=\phi_i\varphi_i$ is product of two continuous functions therefore continuous. + +On $X-\operatorname{supp}(\phi_i)$, $h_i=0$ is continuous. + +By pasting lemma, $h_i$ is continuous. + +Define + +$$ +F: X\to (\mathbb{R}^m\times \mathbb{R})^n +$$ + +where $x\mapsto (h_1(x),\varphi_1(x),h_2(x),\varphi_2(x),\dots,h_n(x),\varphi_n(x))$ + +We want to show that $F$ is imbedding map. + +**(a). $F$ is continuous** + +since it is a product of continuous functions. + +**(b). $F$ is injective** + +that is, if $F(x_1)=F(x_2)$, then $x_1=x_2$. + +By partition of unity, we have, + +$h_1(x_1)=h_1(x_2), h_2(x_1)=h_2(x_2), \dots, h_n(x_1)=h_n(x_2)$. + +And $\varphi_1(x_1)=\varphi_1(x_2), \varphi_2(x_1)=\varphi_2(x_2), \dots, \varphi_n(x_1)=\varphi_n(x_2)$. + +Because $\sum_{i=1}^n \varphi_i(x_1)=1$, therefore the exists $\varphi_i(x_1)=\varphi_i(x_2)>0$. + +Therefore $x1,x_2\in \operatorname{supp}(\phi_i)\subseteq U_i$. + +By definition of $h$, $h_i(x_1)=h_i(x_2)$, $\varphi_i(x_1)\phi_i(x_1)=\varphi_i(x_2)\phi_i(x_2)$. + +Using cancellation, $\phi_i(x_1)=\phi_i(x_2)$. + +Therefore $x_1=x_2$ since $\phi_i(x_1)=\phi_i(x_2)$ is a homeomorphism. + +_In this proof, $\varphi$ ensures the imbedding is properly defined on the open sets_ + +**(c). $F$ is a homeomorphism.** + +Note that by [Theorem 26.6 on Munkres](https://notenextra.trance-0.com/Math4201/Math4201_L25/#theorem-of-closed-maps-from-compact-and-hausdorff-spaces), $F:X\to F(X)$ is a bijective map from a compact space to a Hausdorff space, therefore $F$ is a closed map. + +Since $F$ is continuous, then $F^{-1}(C)$ where $C$ is a closed set in $F(X)$, $F^{-1}(C)$ is closed in $X$. + +Therefore $F$ is a homeomorphism. + +
+ +Then we will prove for the finite partition of unity. + +
+Proof for finite partition of unity + +Some intuitions: + +By definition for partition of unity, consider the sets $W_i,V_i$ defined as + +$$ +W_i=f^{-1}_i((\frac{1}{2n},1])\subseteq f^{-1}_i([\frac{1}{2n},1])\subseteq V_i=f^{-1}_i((0,1])\subseteq \operatorname{supp}(f_i)\subseteq U_i +$$ + +Note that $V_i$ is open and $\overline {V_i}\subseteq U_i$. + +And $\bigcup_{i=1}^n V_i=X$. + +and $W_i$ is open and $\overline{W_i}\subseteq V_i$. + +And $\bigcup_{i=1}^n W_i=X$. + +--- + +Step 1: $\exists$ V_i$ ope subsets $i=1,\dots,n$ such that $\overline{V_i}\subseteq U_i$, and $\bigcup_{i=1}^n V_i=X$. + +For $i=1$, consider $A_1=X-(U_2\cup U_3\cup \dots \cup U_n)$. Therefore $A_1$ is closed, and $A_1\cup U_1=X$. + +So $A_1\subseteq U_1$. + +Note that $A_1$ and $X-U_1$ are disjoint closed subsets of $X$. + +Since $X$ is normal, we can separate disjoint closed subsets $A_1$ and $X-U_1$. + +So we have $A_1\subset V_1\subseteq \overline{V_1}\subseteq U_1$. + +For $i=2$, note that $V_1\cup\left( \bigcup_{i=2}^n U_i\right)=X$, + +Take $A_2=X-\left(V_1\cup\left( \bigcup_{i=3}^n U_i\right)\right)$ (skipping $U_2$). + +Then we have $V_2\subseteq \overline{V_2}\subseteq U_2$. + +For $i=j$, we have + +$$ +A_j=X-\left(\left(\bigcup_{i=1}^{j-1}V_i\right)\cup \left(\bigcup_{i=j+1}^n U_i\right)\right) +$$ + +Continue next lecture. +
diff --git a/content/Math4202/_meta.js b/content/Math4202/_meta.js index b65e509..7bdf740 100644 --- a/content/Math4202/_meta.js +++ b/content/Math4202/_meta.js @@ -7,4 +7,5 @@ export default { Math4202_L2: "Topology II (Lecture 2)", Math4202_L3: "Topology II (Lecture 3)", Math4202_L4: "Topology II (Lecture 4)", + Math4202_L5: "Topology II (Lecture 5)", }