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 Br(p) or B(p; r), is defined by
The closed (metric) ball, which may be denoted by Br 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 Br(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.
Read more about this topic: Ball (mathematics)
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