Metric sphere

A metric sphere in a metric space $(X,d)$ is a set of the form

where $x\in X$ and $r\ge 0$.

A sphere can be written as $\overline{B}_d(x,r)\setminus B_d(x,r)$, so it sits between the open ball and closed ball of the same radius.

Examples:

• In $\mathbb{R}^n$ with the Euclidean metric, $S(x,r)$ is the usual sphere of radius $r$ centered at $x$.
• In $(\mathbb{R},|\cdot|)$, $S(x,r)=\{x-r,x+r\}$ for $r>0$, and $S(x,0)=\{x\}$.