What is homeomorphism?

Question

In which way I define homeomorphism and how can I explain it with the help of example

Answer 1

A homeomorphism is a bijective continuous function with continuous inverse. Please consult literature or Wikipedia for looking up definitions in the fut

An example would be $f: (-\pi/2,\pi/2) \rightarrow \mathbb{R}$, $x \mapsto \operatorname{tan}(x)$ . Can you see why?

Answer 2

Heuristically it's a rubber-sheet transformation, an easy way to see this : Consider the disk $$\mathcal{D}:=\left \{ (x,y) \in \mathbb{R}^2:x^2+y^2 \le r^2, r \in \mathbb{R}_{> 0}\right\}$$ Also consider the ellipse $$\mathcal{E}:=\left\{ (x,y) \in \mathbb{R}^2: \frac{x^2}{a^2}+\frac{y^2}{b^2} \le 1~,~a,b \in \mathbb{R}_{>0}\right\}$$ Here $\mathcal{D}$ and $\mathcal{E}$ are Homeomorphic. The homeomorphism is given by $A: \mathcal{D} \to \mathcal{E}$ $$A(x,y):=r^{-1}\left(ax,by \right)$$ with the inverse operator $$A^{-1}(x,y)=r \left( \frac{x}{a},\frac{y}{b} \right)$$