Open and closed balls - Imported Knowlpedia Content

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 $0<r\le 1$ .