Namespaces
Variants
Actions

Difference between revisions of "Complete metric space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Comment: Completeness is not a topological property, cite Steen & Seebach (1978))
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
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]].
+
{{MSC|54E50}}
 +
{{TEX|done}}
 +
 
 +
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.
 +
 
 +
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 mapping|isometric embedding]], namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$;
 +
*$i(X)$ 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 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 $(-1,1)$. 
 +
 
 +
A [[Compact space|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====
 +
<table>
 +
<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>
 +
</table>

Revision as of 20:51, 9 November 2014

2020 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