Ball (mathematics) - Balls in General Metric Spaces

Balls in General Metric Spaces

Let (M,d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by B_r(p) or B(p; r), is defined by

The closed (metric) ball, which may be denoted by B_r or B, is defined by

Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.

The closure of the open ball B_r(p) is usually denoted . While it is always the case that and, it is not always the case that . For example, in a metric space with the discrete metric, one has and, for any .

An (open or closed) unit ball is a ball of radius 1.

A subset of a metric space is bounded if it is contained in some ball. A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius.

The open balls of a metric space are a basis for a topological space, whose open sets are all possible unions of open balls. This space is called the topology induced by the metric d.