# Compactification

• Nov 15th 2010, 10:49 AM
bram kierkels
Compactification
I want to prove that the one-point compactification of $\mathbb{R}^n$ is homeomorphic to the $n$-sphere

$S^n=\{(y_1,\ldots,y_{n+1})\in\mathbb{R}^{n+1}:y_1^ 2+\cdots +y_{n+1}^2=1\}$.

Thanks
• Nov 15th 2010, 10:56 AM
roninpro
I'm not entirely familiar with compactification, but did you try using the stereographic projection as a homeomorphism?

Stereographic projection - Wikipedia, the free encyclopedia
• Nov 15th 2010, 02:53 PM
Drexel28
Quote:

Originally Posted by bram kierkels
I want to prove that the one-point compactification of $\mathbb{R}^n$ is homeomorphic to the $n$-sphere

$S^n=\{(y_1,\ldots,y_{n+1})\in\mathbb{R}^{n+1}:y_1^ 2+\cdots +y_{n+1}^2=1\}$.

Thanks

Use the fact that $\mathbb{R}^n_{\infty}$ is the unique, up to homeomorphism, space which is compact, Hausdorff, and contains a homeomorphic image of $\mathbb{R}^n$, and the complement of that image is a single point.

Otherwise (if you don't know the above theorem), use roninpro's solution and map $(1,0,\cdots,0)\mapsto\infty$