Compactification

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November 15th 2010, 09:49 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
November 15th 2010, 09:56 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
November 15th 2010, 01:53 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 $(1,0,\cdots,0)\mapsto\infty$

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