Corollary

Every Euclidean sphere is a compact set and is the frontier of the open and closed balls with the same centre and the same radius.

The mapping $x \to \frac{1}{r}(x - x_0)$ transforms the sphere (resp. open ball, closed ball) with centre $x_0$ and radius $r$ into the sphere (resp. open ball, closed ball) with centre $o$ and radius $1$; this sphere is denoted by $S_{n-1}$ and is called the unit sphere in $\mathbf{R}^n$. Likewise, the closed ball with centre $o$ and radius $1$ is denoted by $B_n$ and is called the unit ball in $\mathbf{R}^n$. The topological study of a sphere of $(n - 1)$ dimensions (resp. a closed ball of $n$ dimensions) is thus reduced to that of $S_{n-1}$ (resp. $B_n$). For the open balls, we have the following proposition: