One Point Compactification of $\mathbf{R}^n-\{0\}$

Question

I wanted to know the one point compactification of $\mathbf{R}^n-\{0\}$. The actual problem asks to show that $S^n/\{p,q\} \simeq S^n \vee S^1$, where $p,q$ are two distinct points of $S^n$. I know that $S^n-\{p,q\}$ is homeomorphic to the one point compactification of $\mathbf{R}^n-\{0\}$. So, knowing the result might help to solve the problem. Intuition from $\mathbf{R}-\{0\}$ is not helping much as that is $S^1\vee S^1$ and I couldn't work out for even $n=2$. Please help.

Answer 1

By stereographic projection, you can turn $\Bbb R^n-\{0\}$ into $S^n-\{N,S\}$. You want to make this compact by adding a single point. We wish to add $N$ to fix problems near the north pole and $S$ for problems near the south pole. So to solve both problems at once, we "pinch" $N$ and $S$ together and then only add a single point...