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Difference between revisions of "Complete metric space"

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A metric space in which each [[Cauchy criteria|Cauchy sequence]] converges. A complete metric space is a particular case of a [[Complete uniform space|complete uniform space]].
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A metric space in which each [[Cauchy criteria|Cauchy sequence]] converges. A complete metric space is a particular case of a [[Complete uniform space|complete uniform space]]. A closed subset $A$ of a complete metric $(X,d)$ space is itself a complete metric space (with the distance which is restiction of $d$ to $A$). The converse is true in a general metric space: if $(X,d)$ is a metric space, not necessarily complete, and $A\subset X$ is such that $(A,d)$ is complete, then $A$ is necessarily a closed subset.
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Given any metric space $(X,d)$ there exists a unique [[Completion|completion]] of $X$, that is a triple $(Y,\rho,i)$ such that:
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*$(Y, \rho)$ is a complete metric space;
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*$i: X \to Y$ is an [[Isometric mapping|isometric embedding]], namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$;
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*$i(X)$ is dense in $Y$.
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Often people refer to the metric space $(Y, \rho)$ as the completion. Both the space and the isometric embedding are unique up to isometries.

Revision as of 17:40, 9 December 2013

2010 Mathematics Subject Classification: Primary: 54E50 [MSN][ZBL]

A metric space in which each Cauchy sequence converges. A complete metric space is a particular case of a complete uniform space. A closed subset $A$ of a complete metric $(X,d)$ space is itself a complete metric space (with the distance which is restiction of $d$ to $A$). The converse is true in a general metric space: if $(X,d)$ is a metric space, not necessarily complete, and $A\subset X$ is such that $(A,d)$ is complete, then $A$ is necessarily a closed subset.

Given any metric space $(X,d)$ there exists a unique completion of $X$, that is a triple $(Y,\rho,i)$ such that:

  • $(Y, \rho)$ is a complete metric space;
  • $i: X \to Y$ is an isometric embedding, namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$;
  • $i(X)$ is dense in $Y$.

Often people refer to the metric space $(Y, \rho)$ as the completion. Both the space and the isometric embedding are unique up to isometries.

How to Cite This Entry:
Complete metric space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_metric_space&oldid=30869
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article