Open Ball

An open ball with center and radius in a metric space is the set

It is an open set in the metric topology, and all open balls form a topological basis for that topology (Lebl 2026, pp. 264-266). The same definition applies to balls for a pseudometric.

An -dimensional open ball in Euclidean space consists of the points whose Euclidean distance from its center is less than its radius. Explicitly, with center and radius , it is

The open ball for is called an open interval, and the term open disk is sometimes used for and sometimes as a synonym for open ball.

The ambient metric space matters. In with the Euclidean distance restricted from , the open ball about 0 of radius is , whereas in it is (Lebl 2026, p. 264).