Homeomorphism

Important

This concept is different from Homomorphism

Definition

A homeomorphism between spaces $X,Y$ is a continuous function $\phi :X\to Y$ which has a continuous inverse ${\phi }^{-1}:Y\to X$

necessary but not sufficient condition: $\phi$ is bijective

Warning

Note that $\phi$ is bijective does not mean ${\phi }^{-1}$ is also continuous

In this case $X$ and $Y$ are homeomorphic and denote them as $X\cong Y$ We use the symbol $\cong$ for isomorphism here because a homeomorphism is an isomorphism in the category of topological spaces. (an equivalence relation)

Therefore, the composition of two homeomorphisms is also a homeomorphism, and all self-homeomorphisms $X\to X$ form a group, called the homeomorphism group of $X$, usually denoted by $Homeo(X)$.

Title

所有对一块橡皮泥做的那些不撕不粘的变形 构成了一个特殊的置换群

Properties

Example

Claim: Any two open balls in ${\mathbb{R}}^n$ are homeomorphic we find two functions $t:x\to x+a$ for translation and $d:x\to cx,c\in \mathbb{R}\setminus \{0\}$ for scalation These functions and there inverse are continuous These means $d\circ t$ is a homeomorphism

Claim: The unit ball ${\mathbb{B}}^n$ is homeomorphic to ${\mathbb{R}}^n$ (the ball is open) (${\mathbb{B}}^2$ is the “interior” of a circle) $f:{\mathbb{B}}^n\to {\mathbb{R}}^n$ $x\to \frac{x}{1-\Vert x\Vert }$

$f^{-1}:{\mathbb{R}}^n\to {\mathbb{B}}^n$ $x\to \frac{x}{1+\Vert x\Vert }$

Claim: The unit cube $C=\{(x,y,z)|max(|x|,|y|,|z|)=1\}$ is homeomorphic to ${\mathbb{S}}^2$ (${\mathbb{B}}^n$ is the surface of a sphere in ${\mathbb{R}}^{n+1}$)

Topological property

homomorphism does not preserver size, corners or boundedness

Definition: local homeomorphism

(a weakening of the concept homeomorphism) A map $f:X\to Y$ is a local homeomorphism if each point $x\in X$ has a neighborhood $U\subseteq X$ such that $f(U)$ is open and $f_{|U}:U\to f(U)$ is a homeomorphism

Example

Consider $f:\mathbb{R}\to {\mathbb{S}}^1$ with $x\mapsto e^{i2\pi x}$ Although $f$ is not a homeomorphism ($f$ is not bijective), we can find for any point a neighborhood such that $f$ in this neighborhood is homeomorphic.

Property

A bijective local homeomorphism is a homeomorphism. Proof $U\subseteq X$ open. Choose any $y\in f(U)$. Write $y=f(x)$ for $x\in U$. $x$ has a neighborhood $V_x$ (open) such that $f(V_x)$ open and $f_{|V_x}$ is a homeomorphism. $⟹U\cap V_x$ open (open sets are closed under finite intersections) and $y\in \underset{open}{\underbrace{f(U\cap V_x)}}\subseteq f(U)$ $f(U\cap V_x)$ is open, because $U\cap V_x\subseteq V_x$ is open and $f_{|V_x}$ is homeomorphism $⟹$ any $y\in f(U)$ is contained in a open set. $⟹f(U)$ open $⟹$ $f$ open $⟹$ $f$ is a homeomorphism