Editing Heine–Borel theorem (section)

== Heine–Borel property ==
The Heine–Borel theorem does not hold as stated for general [[Metric space|metric]] and [[topological vector space]]s, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. These spaces are said to have the '''Heine–Borel property'''.

===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}}

===In the theory of topological vector spaces===
A [[topological vector space]] <math>X</math> is said to have the '''Heine–Borel property'''{{sfn|Kirillov|Gvishiani|1982|loc=Theorem 28}} (R.E. Edwards uses the term ''boundedly compact space''{{sfn|Edwards|1965|loc=8.4.7}}) if each closed bounded<ref>A set <math>B</math> in a topological vector space <math>X</math> is said to be ''bounded'' if for each neighborhood of zero <math>U</math> in <math>X</math> there exists a scalar <math>\lambda</math> such that <math>B\subseteq\lambda\cdot U</math>.</ref> set in <math>X</math> is compact.<ref>In the case when the topology of a topological vector space <math>X</math> is generated by some metric <math>d</math> this definition is not equivalent to the definition of the Heine–Borel property of <math>X</math> as a metric space, since the notion of bounded set in <math>X</math> as a metric space is different from the notion of bounded set in <math>X</math> as a topological vector space. For instance, the space <math>{\mathcal C}^\infty[0,1]</math> of smooth functions on the interval <math>[0,1]</math> with the metric <math>d(x,y)=\sum_{k=0}^\infty\frac{1}{2^k}\cdot\frac{\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}{1+\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}</math> (here <math>x^{(k)}</math> is the <math>k</math>-th derivative of the function <math>x\in {\mathcal C}^\infty[0,1]</math>) has the Heine–Borel property as a topological vector space but not as a metric space.</ref> No infinite-dimensional [[Banach space]]s have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional [[Fréchet space]]s do have, for instance, the space <math>C^\infty(\Omega)</math> of smooth functions on an open set <math>\Omega\subset\mathbb{R}^n</math>{{sfn|Edwards|1965|loc=8.4.7}} and the space <math>H(\Omega)</math> of holomorphic functions on an open set <math>\Omega\subset\mathbb{C}^n</math>.{{sfn|Edwards|1965|loc=8.4.7}}  More generally, any quasi-complete [[nuclear space]] has the Heine–Borel property. All [[Montel space]]s have the Heine–Borel property as well.