Editing Complete metric space (section)

==Some theorems==
Every [[Compact space#Metric spaces|compact metric space]] is complete, though complete spaces need not be compact. In fact, a metric space is compact [[if and only if]] it is complete and [[totally bounded]]. This is a generalization of the [[Heine–Borel theorem]], which states that any closed and bounded subspace <math>S</math> of {{math|'''R'''<sup>''n''</sup>}} is compact and therefore complete.<ref>{{cite book |title=Introduction to Metric and Topological Spaces |first=Wilson A. |last=Sutherland|year=1975 |publisher=Clarendon Press |author-link= Wilson Sutherland |isbn=978-0-19-853161-6 }}</ref>

Let <math>(X, d)</math> be a complete metric space. If <math>A \subseteq X</math> is a closed set, then <math>A</math> is also complete. Let <math>(X, d)</math> be a metric space. If <math>A \subseteq X</math> is a complete subspace, then <math>A</math> is also closed.

If <math>X</math> is a [[set (mathematics)|set]] and <math>M</math> is a complete metric space, then the set <math>B(X, M)</math> of all bounded functions {{mvar|f}} from {{mvar|X}} to <math>M</math> is a complete metric space. Here we define the distance in <math>B(X, M)</math> in terms of the distance in <math>M</math> with the [[supremum norm]]
<math display=block>d(f, g) \equiv \sup\{d[f(x), g(x)]: x \in X\}</math>

If <math>X</math> is a [[topological space]] and <math>M</math> is a complete metric space, then the set <math>C_b(X, M)</math> consisting of all [[Continuous function (topology)|continuous]] bounded functions <math>f : X \to M</math> is a closed subspace of <math>B(X, M)</math> and hence also complete.

The [[Baire category theorem]] says that every complete metric space is a [[Baire space]]. That is, the [[union (set theory)|union]] of countably many [[nowhere dense]] subsets of the space has empty [[Interior (topology)|interior]].

The [[Banach fixed-point theorem]] states that a [[contraction mapping]] on a complete metric space admits a [[fixed point (mathematics)|fixed point]]. The fixed-point theorem is often used to [[mathematical proof|prove]] the [[inverse function theorem]] on complete metric spaces such as Banach spaces.

{{Math theorem|name=Theorem<ref name="Zalinescu 2002 p. 33">{{cite book|last=Zalinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific|publication-place=River Edge, N.J. London|year=2002|isbn=981-238-067-1|oclc=285163112|page=33}}</ref>|note=C. Ursescu|math_statement=
Let <math>X</math> be a complete metric space and let <math>S_1, S_2, \ldots</math> be a sequence of subsets of <math>X.</math>

* If each <math>S_i</math> is closed in <math>X</math> then <math display=inline>\operatorname{cl} \left(\bigcup_{i \in \N} \operatorname{int} S_i\right) = \operatorname{cl} \operatorname{int} \left(\bigcup_{i \in \N} S_i\right).</math>
* If each <math>S_i</math> is [[open subset|open]] in <math>X</math> then <math display=inline>\operatorname{int} \left(\bigcap_{i \in \N} \operatorname{cl} S_i\right) = \operatorname{int} \operatorname{cl} \left(\bigcap_{i \in \N} S_i\right).</math>
}}