Proofs involving stereographic projection

Question

Can someone please help me finding references in which one has used following homeomorphism to solve general topology problems? $S\colon S_{0}^{n}\to \mathbb{R}^n$ is defined by for any $x=(x_1,...,x_{n+1})$, $$ S(x) = \left(\frac{x_1}{1-x_{n+1}},\ldots,\frac{x_n}{1-x_{n+1}} \right) ,$$ where $S_{0}^{n}$ is the $n$-sphere in $\mathbb{R}^{n+1}$ with north pole $(0,0,\ldots,1)$ deleted. Note that $S$ is just a stereographic projection.

Answer 1

One can use the stereographic projection to show that the sphere $\mathbb{S}^n$ is homeomorphic to the one point compactification of $\mathbb{R}^n$ for $n \geq1.$