I want to prove that the one-point compactification of $\displaystyle \mathbb{R}^n$ is homeomorphic to the $\displaystyle n$-sphere
$\displaystyle S^n=\{(y_1,\ldots,y_{n+1})\in\mathbb{R}^{n+1}:y_1^ 2+\cdots +y_{n+1}^2=1\}$.
Thanks
I'm not entirely familiar with compactification, but did you try using the stereographic projection as a homeomorphism?
Stereographic projection - Wikipedia, the free encyclopedia
Use the fact that $\displaystyle \mathbb{R}^n_{\infty}$ is the unique, up to homeomorphism, space which is compact, Hausdorff, and contains a homeomorphic image of $\displaystyle \mathbb{R}^n$, 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 $\displaystyle (1,0,\cdots,0)\mapsto\infty$