One-point compactification. The local compactness and para compactness is the concept but I unable to understand the basic thought of this problem

Question

Prove that the one-point compactification of $R^n$ is homeomorphic to $S^n$.

Answer 1

It’s well-known that the stereographic projection shows that $\Bbb S^n\setminus \{p\} \simeq \Bbb R^n$ (where $p$ is the north pole of the sphere, typically) and then apply

Theorem If $X$ is compact Hausdorff and $Y$ is locally compact Hausdorff and for some point $p \in X$ we have that $Y \simeq X\setminus \{p\}$, then $X$ is homeomorphic to the one-point compactification of $Y$.