I'm looking for a reference to rigorous proofs of the following two claims (if someone is willing to write down a proof that would also be excellent):
- The Stereographic Projection is a Homeomorphism between $S^{n}\backslash\left\{ N\right\}$ (the sphere without its north pole) and $\mathbb{R}^{n}$ for $n\geq2$.
- The Stereographic Projection is a Homeomorphism between $S^{n}$ and the one point compactification of $\mathbb{R}^{n}$
Help would be appreciated.
For the first request just try to write down explicitly the function that defines such a projection, by considering an hyperplane which cuts the sphere along the equator. Consider $S^n$ in $R^{n+1}$, with $R^n$ as the subset with $x_{n+1}=0$. The North pole is $(0,0,..,0,1)$ and the image of each point is the intersection of the line between such a point and the north pole and the above mentioned hyperplane. Thus you need to find (solving with respect to $t$) $\{(0,...,1) + t((x_1,..,x_{n+1})-(0,...,1)): t \in \mathbb{R} \}\bigcap\{x_{n+1}=0\}$ which yields the desired $t$ and so the image of the point.