From c42e7e6489f6e86a9419a408765669f15f6263ab Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Fri, 5 Dec 2025 11:53:00 -0600 Subject: [PATCH] updates --- content/Math4201/Math4201_L35.md | 2 +- content/Math4201/Math4201_L36.md | 4 +- content/Math4201/Math4201_L37.md | 4 +- content/Math4201/Math4201_L38.md | 4 +- content/Math4201/Math4201_L39.md | 120 +++++++++++++++++++++++++++++++ content/Math4201/_meta.js | 1 + 6 files changed, 128 insertions(+), 7 deletions(-) create mode 100644 content/Math4201/Math4201_L39.md diff --git a/content/Math4201/Math4201_L35.md b/content/Math4201/Math4201_L35.md index 8cf210f..f0b9fbf 100644 --- a/content/Math4201/Math4201_L35.md +++ b/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. diff --git a/content/Math4201/Math4201_L36.md b/content/Math4201/Math4201_L36.md index 93334ca..bb3411d 100644 --- a/content/Math4201/Math4201_L36.md +++ b/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 diff --git a/content/Math4201/Math4201_L37.md b/content/Math4201/Math4201_L37.md index a1e4ffe..9a1cb08 100644 --- a/content/Math4201/Math4201_L37.md +++ b/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 diff --git a/content/Math4201/Math4201_L38.md b/content/Math4201/Math4201_L38.md index 5860b5c..931688e 100644 --- a/content/Math4201/Math4201_L38.md +++ b/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.
diff --git a/content/Math4201/Math4201_L39.md b/content/Math4201/Math4201_L39.md new file mode 100644 index 0000000..645569d --- /dev/null +++ b/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) + +
+ Example of manifold + +$\mathbb{R}^d$ 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. + +
+ +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. + +
+Proof for embedding compact manifolds + +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. + +
\ No newline at end of file diff --git a/content/Math4201/_meta.js b/content/Math4201/_meta.js index 914cb15..7684633 100644 --- a/content/Math4201/_meta.js +++ b/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)", }