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Difference between revisions of "Complete metric space"

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(Comment: Completeness is not a topological property, cite Steen & Seebach (1978))
 
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A metric space in which each fundamental, or Cauchy, sequence converges. A complete metric space is a particular case of a [[Complete uniform space|complete uniform space]].
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{{MSC|54E50}}
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{{TEX|done}}
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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]]. A closed subset $A$ of a complete metric $(X,d)$ space is itself a complete metric space (with the distance which is the restriction 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]] 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.
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====Comments====
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Completeness is not a [[Topological invariant|topological property]], that is, there are metric spaces which are [[Topological equivalence|homeomorphic]] as topological spaces, one being complete and the other not.  For example, consider the real line $\mathbb{R}$ and the open unit interval $(-1,1)$, each with the usual metric.  There are homeomorphic, for example via the map $x \mapsto x / (1 + |x|)$.  However, as metric spaces, $\mathbb{R}$ is complete, but the sequence $1-1/n$ is a Cauchy sequence which does not converge in $(-1,1)$. 
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A [[Compact space|compact]] metric space is complete.
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A topological space is ''topologically complete'' if there is a complete metric space structure compatible with the given topology: this is a topological property.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Steen, Lynn Arthur; Seebach, J. Arthur Jr., ''Counterexamples in Topology'' (second edition).  Springer-Verlag (1978). ISBN 978-0-486-68735-3. MR 507446.  Zbl 0386.54001.</TD></TR>
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</table>

Latest revision as of 20:51, 9 November 2014

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 the restriction 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.

Comments

Completeness is not a topological property, that is, there are metric spaces which are homeomorphic as topological spaces, one being complete and the other not. For example, consider the real line $\mathbb{R}$ and the open unit interval $(-1,1)$, each with the usual metric. There are homeomorphic, for example via the map $x \mapsto x / (1 + |x|)$. However, as metric spaces, $\mathbb{R}$ is complete, but the sequence $1-1/n$ is a Cauchy sequence which does not converge in $(-1,1)$.

A compact metric space is complete.

A topological space is topologically complete if there is a complete metric space structure compatible with the given topology: this is a topological property.

References

[1] Steen, Lynn Arthur; Seebach, J. Arthur Jr., Counterexamples in Topology (second edition). Springer-Verlag (1978). ISBN 978-0-486-68735-3. MR 507446. Zbl 0386.54001.
How to Cite This Entry:
Complete metric space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_metric_space&oldid=13111
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article