Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

Question

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ?

I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point from two spaces in order to be able to distinguish them homologically.

Answer 1

The most basic such map is:

$$x\mapsto \frac x{\|x\|}$$

The homotopy is:

$$h(x,t)=\frac{x}{(1-t)+t\|x\|}$$