updates

content/Math4202/Math4202_L3.md

@@ -75,7 +75,8 @@ The map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient ma
 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 $\mathbb{R}^m$.
+2. With a countable basis
+3. Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
 
 > [!NOTE]
 >

content/Math4202/Math4202_L4.md

@@ -0,0 +1,75 @@
+# Math4202 Topology II (Lecture 4)
+
+## Manifolds
+
+### Imbedding of Manifolds
+
+#### Definition of Manifold
+
+An $m$-dimensional **manifold** is a topological space  $X$ that is
+
+1. Hausdorff
+2. With a countable basis
+3. Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$. (local euclidean)
+
+> [!NOTE]
+>
+> Try to find some example that satisfies some of the properties above but not a manifold.
+
+1. Non-Hausdorff
+2. Non-countable basis
+   - Consider $\mathbb{R}^\delta$ where the set is $\mathbb{R}$ with discrete topology. The basis must include all singleton sets in $\mathbb{R}$ therefore $\mathbb{R}^\delta$ is not second countable.
+3. Non-local euclidean
+   - Consider the subspace topology over segment $[0,1]$ on real line, the subspace topology is not local euclidean since the open set containing the end point $[0,a)$ is not homeomorphic to open sets in $\mathbb{R}$. (if we remove the end point, in the segment space we have $(0,a)$ but in $\mathbb{R}$ is $(-a,0)\cup (0,a)$, which is not connected. Therefore cannot be homeomorphic to open sets in $\mathbb{R}$)
+   - Any shape with intersection is not local euclidean.
+
+#### 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)_
+
+<details>
+<summary>Proof for Whithney's Embedding Theorem</summary>
+
+Since $X$ is a compact manifold, $\forall x\in X$, there is an open neighborhood $U_x$ of $x$ such that $U_x$ is homeomorphic to $\mathbb{R}^d$. 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 existsence of 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.
+
+Continue on next lecture.
+
+</details>

content/Math4202/_meta.js

@@ -6,4 +6,5 @@ export default {
     Math4202_L1: "Topology II (Lecture 1)",
     Math4202_L2: "Topology II (Lecture 2)",
     Math4202_L3: "Topology II (Lecture 3)",
+    Math4202_L4: "Topology II (Lecture 4)",
 }

content/Math4302/Math4302_L4.md

@@ -0,0 +1,2 @@
+# Math4302 Modern Algebra (Lecture 4)
+

content/Math4302/_meta.js

@@ -6,4 +6,5 @@ export default {
     Math4302_L1: "Modern Algebra (Lecture 1)",
     Math4302_L2: "Modern Algebra (Lecture 2)",
     Math4302_L3: "Modern Algebra (Lecture 3)",
+    Math4302_L4: "Modern Algebra (Lecture 4)",
 }