# 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
   - Real line with two origin, as discussed in homework problem
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>