Open and closed balls

Let $(X,d)$ be a metric space , let $x_0\in X$, and let $r>0$.

• The open ball of center $x_0$ and radius $r$ is

  $$B(x_0;r):=\{x\in X\mid d(x,x_0)<r\}.$$

• The closed ball of center $x_0$ and radius $r$ is

  $$B'(x_0;r):=\{x\in X\mid d(x,x_0)\le r\}.$$

Open balls generate the topology of the metric space: a set is open iff it contains an open ball around each of its points.

Examples:

• In $\mathbb{R}$ with $d(x,y)=|x-y|$, $B(x_0;r)=(x_0-r,x_0+r)$ and $B'(x_0;r)=[x_0-r,x_0+r]$.
• In the discrete metric, $B(x_0;r)=\{x_0\}$ whenever $0<r\le 1$.