Editing Heine–Borel theorem (section)

===In the theory of metric spaces=== 
A [[metric space]] <math>(X,d)</math> is said to have the '''Heine–Borel property''' if each closed bounded<ref>A set <math>B</math> in a metric space <math>(X,d)</math> is said to be ''bounded'' if it is contained in a ball of a finite radius, i.e. there exists <math>a\in X</math> and <math>r>0</math> such that <math>B\subseteq\{x\in X\mid d(x,a)\le r\}</math>.</ref> set in <math>X</math> is compact.

Many metric spaces fail to have the Heine–Borel property, such as the metric space of [[rational number]]s (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional [[Banach space]]s have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.

A metric space <math>(X,d)</math> has a Heine–Borel metric which is Cauchy locally identical to <math>d</math> if and only if it is [[complete space|complete]], [[sigma-compact|<math>\sigma</math>-compact]], and [[locally compact space|locally compact]].{{sfn|Williamson|Janos|1987}}