Closed ball

The set of points within a given radius of a center point in a metric space, using non-strict inequality.

Closed ball

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

\overline{B}_d(x,r)=\{y\in X : d(x,y)\le r\},

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

Closed balls are closely related to open balls and are closed sets in the metric-induced topology .

Examples:

In $(\mathbb{R},|\cdot|)$ , $\overline{B}(x,r)=[x-r,x+r]$ .
In the discrete metric on $X$ , $\overline{B}(x,0)=\{x\}$ and $\overline{B}(x,1)=X$ .