Closed Ball

A closed ball with center and radius in a metric space is the set

It is a closed set in the metric topology (Lebl 2026, pp. 264-266). For , it is the singleton set .

In Euclidean space, the closed ball with center and radius is

For , the topological closure of the open ball is contained in , but equality need not hold in a general metric space. For example, let contain at least two points and have the discrete metric, with distance 1 between distinct points. The open ball is already a closed set, whereas (Knapp 2016, p. 98). In Euclidean space, equality with the topological closure does hold for positive radius.