updates

content/Math4201/Math4201_L35.md

@@ -112,7 +112,7 @@ Recall that $\mathbb{R}_{\ell}$ with lower limit topology is normal. But $\mathb
 
 This shows that $\mathbb{R}_{\ell}$ is not metrizable. Otherwise $\mathbb{R}_{\ell}\times \mathbb{R}_{\ell}$ would be metrizable. Which could implies that $\mathbb{R}_{\ell}$ is normal.
 
-#### Theorem of metrizability
+#### Theorem of metrizability (Urysohn metirzation theorem)
 
 If $X$ is normal and second countable, then $X$ is metrizable.
 

content/Math4201/Math4201_L36.md

@@ -1,8 +1,8 @@
 # Math4201 Topology I (Lecture 36)
 
-## Countable Axioms and Separation Axioms
+## Separation Axioms
 
-### Separation Axioms
+### Regular spaces
 
 #### Proposition for $T_1$ spaces
 

content/Math4201/Math4201_L37.md

@@ -1,8 +1,8 @@
 # Math4201 Topology I (Lecture 37)
 
-## Countable Axioms and Separation Axioms
+## Separation Axioms
 
-### Continue on Normal spaces
+### Normal spaces
 
 #### Proposition of normal spaces
 

content/Math4201/Math4201_L38.md

@@ -30,9 +30,9 @@ Choose $N\geq \frac{1}{\epsilon}$, then $\forall n\geq N,\frac{\overline{d}(x_n,
 
 We will use the topological space above to prove the following theorem.
 
-#### Theorem for metrizable spaces
+#### Urysohn metrization theorem
 
-If $X$ is a regular and second countable topological space, then $X$ is metrizable.
+If $X$ is a normal (regular and second countable) topological space, then $X$ is metrizable.
 
 <details>
 

content/Math4201/Math4201_L39.md

@@ -0,0 +1,120 @@
+# Math4201 Topology I (Lecture 39)
+
+## Separation Axioms
+
+### Embedding manifolds
+
+A $d$ dimensional manifold is the topological space satisfying the following three properties:
+
+1. Haudorff property ($\forall x,y\in X, \exists U,V\in \mathcal{T}_X$ such that $x\in U\cap V$ and $y\notin U\cap V$)
+2. Second countable property ($\exists \mathcal{B}\subseteq \mathcal{T}_X$ such that $\mathcal{B}$ is a basis for $X$ and $\mathcal{B}$ is countable)
+3. Local homeomorphism to $\mathbb{R}^d$ ($\forall x\in M$, there is a neighborhood $U$ of $x$ such that $U$ is homeomorphic to $\mathbb{R}^d$. $\varphi:U\to \mathbb{R}^d$ is bijective, continuous, and open)
+
+<details>
+<summary> Example of manifold</summary>
+
+$\mathbb{R}^d$ is a $d$-dimensional manifold. And any open subspace of $\mathbb{R}^d$ is also a manifold.
+
+---
+
+$S^1$ is a $1$-dimensional manifold.
+
+---
+
+$T=\mathbb{R}^2/\mathbb{Z}^2$ is a $2$-dimensional manifold.
+
+</details>
+
+Recall the [Urysohn metirzation theorem](./Math4201_L38.md/#urysohn-metirzation-theorem). Any normal and second countable space is metrizable.
+
+In the proof we saw that any such space can be embedded into $\mathbb{R}^\omega$ with the product topology.
+
+Question: What topological space can be embedded into $\mathbb{R}^n$ with the product topology?
+
+#### Theorem for embedding compact manifolds into $\mathbb{R}^n$
+
+Any $d$-dimensional (compact, this assumption makes the proof easier) manifold can be embedded into $\mathbb{R}^n$ with the product topology.
+
+#### Definition for support of function
+
+$\operatorname{supp}(f)=f^{-1}(\mathbb{R}-\{0\})$
+
+#### Definition for partition of unity
+
+Let $\{U_i\}_{i=1}^n$ be an open covering of $X$. A partition of unity for $X$ dominated by $\{U_i\}_{i=1}^n$ is a set of functions $\phi_i:X\to\mathbb{R}$ such that:
+
+1. $\operatorname{supp}(\phi_i)\subseteq U_i$
+2. $\sum_{i=1}^n \phi_i(x)=1$ for all $x\in X$
+
+#### Theorem for existence of partition of unity
+
+Let $X$ be a normal space and $\{U_i\}_{i=1}^n$ is an open covering of $X$. Then there is a partition of unity dominated by $\{U_i\}_{i=1}^n$.
+
+Proof uses Urysohn's lemma.
+
+<details>
+<summary>Proof for embedding compact manifolds</summary>
+
+Let $M$ be a compact manifold.
+
+For any point $x\in M$, there is an open neighborhood $U_x$ of $x$ such that $U_x$ is homeomorphic to $\mathbb{R}^d$.
+
+Let $\{U_x\}_{x\in M}$ be an open cover of $M$.
+
+Since $M$ is compact, $\{U_x\}_{x\in M}$ has a finite subcover.
+
+then $\{U_{x_i}\}_{i=1}^n$ is an open cover of $M$.
+
+Therefore $F_i:U_{x_i}\to \mathbb{R}^d$ is a homeomorphism.
+
+Since $M$ is compact and second countable, $M$ is normal.
+
+Then there sis a partition of unity $\{\phi_i:X\to \mathbb{R}\}_{i=1}^n$ for $M$ with support by $\{U_{x_i}\}_{i=1}^n$ dominated by $\{U_{x_i}\}_{i=1}^n$. Where
+
+- $\sum_{i=1}^n \phi_i(x)=1$
+- $\operatorname{supp}(\phi_i)\subseteq U_{x_i}$
+
+Define $\Psi:X\to \mathbb{R}^d$ as
+
+$$
+\Psi_i(x)=\begin{cases}
+\phi_i(x)F_i(x) & \text{if } x\in U_{x_i} \\
+0 & x\in X-\operatorname{supp}(\phi_i)
+\end{cases}
+$$
+
+Note that $\operatorname{supp}(\phi_i)\subseteq U_{x_i}$, this implies that $(X-\operatorname{supp}(\phi_i))\cup U_{x_i}=X$.
+
+$U_{x_i}\cap (X-\operatorname{supp}(\phi_i))= U_i-\operatorname{supp}(\phi_i)$
+
+In particualr, for any $x$ in the intersection, $\phi_i(x)=0\implies \phi_i(x)F_i(x)=0$.
+
+So on the overlap, $\phi_i(x)F_i(x)=0$ and hence $\Psi_i$ is well defined.
+
+Define $\Phi:X\to \mathbb{R}\times \dots \times \mathbb{R}\times \mathbb{R}^d\times \dots \times \mathbb{R}^d\cong \mathbb{R}^{(1+d)n}$ as
+
+$$
+\Phi(x)=(\phi_1(x),\dots,\phi_n(x),\Psi_1(x),\dots,\Psi_n(x))
+$$
+
+This is continuous because $\phi_i(x)$ and $\Psi_i(x)$ are continuous.
+
+Since $M$ is compact, we just need to show that $\Phi$ is one-to-one to verify that it is an embedding.
+
+Let $\Phi(x)=\Phi(x')$, then $\forall i,\phi_i(x)=\phi_i(x')$, and $\forall i,\Psi_i(x)=\Psi_i(x')$.
+
+Since $\sum_{i=1}^n \phi_i(x)=1$, $\exists i$ such that $\phi_i(x)\neq 0$, therefore $x\in U_{x_i}$.
+
+Since $\phi_i(x)=\phi_i(x')$, then $x'\in U_{x_i}$.
+
+This implies that $\Psi_i(x)=\Psi_i(x')$, $\phi_i(x)F_i(x)=\phi_i(x')F_i(x')$.
+
+So $F_i(x)=F_i(x')$ since $F_i$ is a homeomorphism.
+
+This implies that $x=x'$.
+
+So $\Phi$ is one-to-one, it is injective.
+
+Therefore $\Phi$ is an embedding.
+
+</details>

content/Math4201/_meta.js

@@ -42,4 +42,5 @@ export default {
     Math4201_L36: "Topology I (Lecture 36)",
     Math4201_L37: "Topology I (Lecture 37)",
     Math4201_L38: "Topology I (Lecture 38)",
+    Math4201_L39: "Topology I (Lecture 39)",
 }