# Math4202 Topology II (Lecture 3)

## Reviewing quotient map

### Quotient map from equivalence relation

Consider $X,Y$ be two topological space and $A\subset X$, where $f:A\to Y$ is a function.

Then the disjoint union $X\sqcup Y /_{a\sim f(a)}$ is a quotient space of $X\sqcup Y$ by the equivalence relation $a\sim f(a)$

Consider $e^n$ be the n dimensional closed ball (n-cells)

$$
e^n=\{x\in \mathbb{R}^n:\sum_{i=1}^n x_i^2\leq 1\}
$$

and $\partial e^n=A$ be the $n-1$ dimensional sphere.

#### CW complex

Let $X_0$ be arbitrary set of points.

Then we can create $X_1$ by

$$
X_1=\{(e_\alpha^1,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^1\to X_0\}
$$

where $\varphi$ is a continuous map, and $e_\alpha^1$ is a $1$-cell (interval).

$$
X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
$$

and $e_\alpha^2$ is a $2$-cell (disk). (mapping boundary of disk to arc (like a croissant shape, if you try to preserve the area))

The higher dimensional folding cannot be visualized in 3D space.

$$
X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1}
$$

<details>
<summary>Example of CW complex construction</summary>

$X_0=a$

$X_1=$ circle, with end point and start point at $a$

$X_2=$ sphere (shell only), with boundary shrinking at the circle create by $X_1$

---

$X_0=a$

$X_1=a$

$X_2=$ ballon shape with boundary of circle collapsing at $a$
</details>

#### Theorem of quotient space

Let $p:X\to Y$ be a quotient map, let $Z$ be a space and $g:X\to Z$ be a map that is constant on each set $p^{-1}(y)$ for each $y\in Y$.

Then $g$ induces a map $f: X\to Z$ such that $f\circ p=g$.

The map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient map if and only if $g$ is a quotient map.

## Imbedding of Manifolds

### Manifold

#### 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 $\mathbb{R}^m$.

> [!NOTE]
>
> Try to find some example that satisfies some of the properties above but not a manifold.