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}

Example of CW complex construction

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.