Editing Ball (mathematics) (section)

==In general metric spaces==
Let {{math|(''M'', ''d'')}} be a [[metric space]], namely a set {{mvar|M}} with a [[Metric (mathematics)|metric]] (distance function) {{mvar|d}}, and let {{tmath|r}} be a positive real number. The open (metric) '''ball of radius''' {{mvar|r}} centered at a point {{mvar|p}} in {{mvar|M}}, usually denoted by {{math|''B<sub>r</sub>''(''p'')}} or {{math|''B''(''p''; ''r'')}}, is defined the same way as a Euclidean ball, as the set of points in {{mvar|M}} of distance less than {{mvar|r}} away from {{mvar|p}},
<math display="block">B_r(p) = \{ x \in M \mid d(x,p) < r \}.</math>

The ''closed'' (metric) ball, sometimes denoted {{math|''B<sub>r</sub>''[''p'']}} or {{math|''B''[''p''; ''r'']}}, is likewise defined as the set of points of distance less than or equal to {{mvar|r}} away from {{mvar|p}},
<math display="block">B_r[p] = \{ x \in M \mid d(x,p) \le r \}.</math>

In particular, a ball (open or closed) always includes {{mvar|p}} itself, since the definition requires {{math|''r'' > 0}}. A '''[[unit ball]]''' (open or closed) is a ball of radius 1.

A ball in a general metric space need not be round. For example, a ball in [[real coordinate space]] under the [[Chebyshev distance]] is a [[hypercube]], and a ball under the [[taxicab distance]] is a [[cross-polytope]]. A closed ball also need not be [[Compact space|compact]]. For example, a closed ball in any infinite-dimensional [[normed vector space]] is never compact. However, a ball in a vector space will always be [[Convex set|convex]] as a consequence of the triangle inequality.

A subset of a metric space is [[bounded set|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]] can serve as a [[base (topology)|base]], giving this space a [[topological space|topology]], the open sets of which are all possible [[union (set theory)|union]]s of open balls. This topology on a metric space is called the '''topology induced by''' the metric {{mvar|d}}.

Let <math>\overline{B_r(p)}</math> denote the [[closure (topology)|closure]] of the open ball <math>B_r(p)</math> in this topology. While it is always the case that <math>B_r(p) \subseteq \overline{B_r(p)} \subseteq B_r[p],</math> it is {{em|not}} always the case that <math>\overline{B_r(p)} = B_r[p].</math> For example, in a metric space <math>X</math> with the [[discrete metric]], one has <math>\overline{B_1(p)} = \{p\}</math> but <math>B_1[p] = X</math> for any <math>p \in X.</math>