Math4202 Topology II (Lecture 3)

Reviewing quotient map

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.

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}

$X_0=a$

$X_1=$ 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 $a$

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.

An $m$ -dimensional manifold is a topological space $X$ that is

Hausdorff
With a countable basis
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.