Proposition 2

Every $n$-dimensional open ball is homeomorphic to $\mathbf{R}^n$.

For the mapping $x \to \frac{x}{1 + ||x||}$ is continuous on $\mathbf{R}^n$ and maps $\mathbf{R}^n$ onto the open ball with centre $o$ and radius $1$; moreover, from $y = \frac{x}{1 + ||x||}$ we deduce $x = \frac{y}{1 - ||y||}$, so that the mapping is bijective and bicontinuous.

Let $\mathbf{R}_n^*$ denote the complement of $o$ in $\mathbf{R}^n$.