Closed ball

Let $(X,d)$ be a metric space , let $x\in X$, and let $r\ge 0$. The closed ball of radius $r$ centered at $x$ is

Closed balls are typically closed sets in metric spaces and are used to describe boundedness and compactness phenomena (e.g., bounded sets lie inside some closed ball). Compare with open ball .

Examples:

• In $\mathbb{R}$, $\overline{B}(a,r)=[a-r,a+r]$.
• In $\mathbb{R}^2$, $\overline{B}(0,1)$ is the closed unit disk.
• In a discrete metric space, $\overline{B}(x,1)=X$ if the metric is $0/1$ (since $d(x,y)\le 1$ for all $y$).