Editing Metric space (section)

===Completeness===
{{main|Complete metric space}}
Informally, a metric space is ''complete'' if it has no "missing points": every sequence that looks like it should converge to something actually converges.

To make this precise: a sequence {{math|(''x<sub>n</sub>'')}} in a metric space {{mvar|M}} is [[Cauchy sequence|''Cauchy'']] if for every {{math|ε > 0}} there is an integer {{mvar|N}} such that for all {{math|''m'', ''n'' > ''N''}}, {{math|''d''(''x<sub>m</sub>'', ''x<sub>n</sub>'') < ε}}.  By the triangle inequality, any convergent sequence is Cauchy: if {{mvar|x<sub>m</sub>}} and {{mvar|x<sub>n</sub>}} are both less than {{math|ε}} away from the limit, then they are less than {{math|2ε}} away from each other.  If the converse is true—every Cauchy sequence in {{mvar|M}} converges—then {{mvar|M}} is complete.

Euclidean spaces are complete, as is <math>\R^2</math> with the other metrics described above.  Two examples of spaces which are not complete are {{open-open|0, 1}} and the rationals, each with the metric induced from <math>\R</math>.  One can think of {{open-open|0, 1}} as "missing" its endpoints 0 and 1.  The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in <math>\R</math> (for example, its successive decimal approximations).  These examples show that completeness is ''not'' a topological property, since <math>\R</math> is complete but the homeomorphic space {{open-open|0, 1}} is not.

This notion of "missing points" can be made precise.  In fact, every metric space has a unique [[completion (metric space)|''completion'']], which is a complete space that contains the given space as a [[dense set|dense]] subset. For example, {{closed-closed|0, 1}} is the completion of {{open-open|0, 1}}, and the real numbers are the completion of the rationals.

Since complete spaces are generally easier to work with, completions are important throughout mathematics.  For example, in abstract algebra, the [[p-adic numbers|''p''-adic numbers]] are defined as the completion of the rationals under a different metric.  Completion is particularly common as a tool in [[functional analysis]].  Often one has a set of nice functions and a way of measuring distances between them.  Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways.  For example, [[weak solution]]s to [[differential equation]]s typically live in a completion (a [[Sobolev space]]) rather than the original space of nice functions for which the differential equation actually makes sense.
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If <math>X</math> is a complete subset of the metric space <math>M</math>, then <math>X</math> is closed in <math>M</math>. Indeed, a space is complete if and only if it is closed in any containing metric space.

Every complete metric space is a [[Baire space]].
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