Homeomorphism between a cylinder and a plane?

[Fernando Revilla]

I quote a question from Yahoo! Answers

Because there is a homeomorphism between a cylinder and a plane?

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

We can construct an explicit homeomorphism between the punctured plane (for example $\mathbb{R}^2\setminus\{(0,0)\}$) and a cylinder (for example $C=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1\}$) given by:
$$f:\mathbb{R}^2\setminus\{(0,0)\}\to C\;,\quad f(r\cos\theta,r\sin\theta)=(\cos \theta,\sin\theta,\ln r)\;(r>0)$$
But $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2\setminus\{(0,0)\}$, because simply connected is a toplogical property.