# Showing that the open ball is homeomorphic to $\mathbb{R}^n$ [duplicate]

I'm trying to prove that an open ball is homeomorphic to $\mathbb{R}^n$ but since this is the first proof of this kind that I try to give I'm having a little doubt on how to begin. I've had some thoughts about working with the inclusion map, however I'm not sure if I'm on the right way. Can someone give just some hint on how to start this proof ?

Thanks for your help in advance, and sorry if the question is too basic.

## marked as duplicate by Cameron Buie, Joe, Micah, Noah Snyder, Stefan HansenApr 9 '13 at 5:02

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• Try constructing an example for $n = 1$. Can you find an explicit homeomorphism between $(0,1)$ and $\mathbb{R}$? – Christopher A. Wong Apr 9 '13 at 1:36

## 1 Answer

Remember you have to regard the open ball as a metric space on its own.

Hint Look at $f:\Bbb R\to (-1,1)$ $$f(x)=\frac{x}{1+|x|}$$

Then use this coordinate-wise.