Proposition 3

The space $\mathbf{R}_n^*$ is homeomorphic to the product of $S_{n-1}$ and the space $\mathbf{R}_+^*$ of real numbers $> 0$.

For every point $x \neq o$ can be written uniquely in the form $tz$, where $t > 0$ and $||z|| = 1$, since $x = tz$ implies $t = ||x||$ and $z = x/||x||$. Since $tz$ is continuous on the product $\mathbf{R} \times \mathbf{R}^n$ and hence a fortiori on $\mathbf{R}_+^* \times S_{n-1}$, and since $||x||$ and $\frac{1}{||x||}$ are continuous on $\mathbf{R}_n^*$, the proposition is proved.

The mapping $x \to x/||x||$ is called the central projection of $\mathbf{R}_n^*$ onto $S_{n-1}$. One defines in the same way the central projection of the complement of a point $a$ onto a sphere with centre $a$.