Complete metric space: Difference between revisions
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In [[mathematics]], ''' | {{subpages}} | ||
In [[mathematics]], a '''complete metric space''' is a [[metric space]] in which every [[Cauchy sequence]] in that space is ''convergent''. In other words, every Cauchy sequence in the metric space tends in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete." | |||
==Formal definition== | ==Formal definition== | ||
Let ''X'' be a metric space with metric ''d''. Then ''X'' is complete if for every Cauchy sequence <math>x_1,x_2,\ldots \in X</math> there is an associated element <math>x \in X</math> such that <math>\mathop{\lim}_{n \rightarrow \infty} d(x_n,x)=0</math>. | Let ''X'' be a metric space with metric ''d''. Then ''X'' is complete if for every Cauchy sequence <math>x_1,x_2,\ldots \in X</math> there is an associated element <math>x \in X</math> such that <math>\mathop{\lim}_{n \rightarrow \infty} d(x_n,x)=0</math>. | ||
== | ==Examples== | ||
[[ | * The real numbers '''R''', and more generally finite-dimensional [[Euclidean space]]s, with the usual metric are complete. | ||
* Any [[compact space|compact]] metric space is [[sequentially compact space|sequentially compact]] and hence complete. The converse does not hold: for example, '''R''' is complete but not compact. | |||
* In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete. | |||
* The rational numbers '''Q''' are ''not'' complete. For example, the sequence (''x''<sub>''n''</sub>) defined by ''x''<sub>0</sub> = 1, ''x''<sub>''n''+1</sub> = 1 + 1/''x''<sub>''n''</sub> is Cauchy, but does not converge in '''Q'''. | |||
==Completion== | |||
Every metric space ''X'' has a '''completion''' <math>\bar X</math> which is a complete metric space in which ''X'' is [[isometry|isometrically]] embedded as a [[Denseness|dense]] subspace. The completion has a universal property. | |||
===Examples=== | |||
* The real numbers '''R''' are the completion of the rational numbers '''Q''' with respect to the usual metric of absolute distance. | |||
==Topologically complete space== | |||
Completeness is not a [[topological property]]: it is possible for a complete metric space to be [[homeomorphism|homeomorphic]] to a metric space which is not complete. For example, the map | |||
:<math> t \leftrightarrow \left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right) </math> | |||
is a homeomorphism between the complete metric space '''R''' and the incomplete space which is the [[unit circle]] in the [[Euclidean plane]] with the point (0,-1) deleted. The latter space is not complete as the non-Cauchy sequence corresponding to ''t''=''n'' as ''n'' runs through the [[positive integer]]s is mapped to a non-convergent Cauchy sequence on the circle. | |||
[[ | We can define a [[topological space]] to be ''metrically topologically complete'' if it is homeomorphic to a complete metric space. A topological condition for this property is that the space be [[metrizable space|metrizable]] and an ''absolute G<sub>δ</sub>'', that is, a [[G-delta set|G<sub>δ</sub>]] in every topological space in which it can be embedded. | ||
[[ | ==See also== | ||
[[Category: | * [[Banach space]] | ||
* [[Hilbert space]][[Category:Suggestion Bot Tag]] | |||
Latest revision as of 11:01, 31 July 2024
In mathematics, a complete metric space is a metric space in which every Cauchy sequence in that space is convergent. In other words, every Cauchy sequence in the metric space tends in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete."
Formal definition
Let X be a metric space with metric d. Then X is complete if for every Cauchy sequence there is an associated element such that .
Examples
- The real numbers R, and more generally finite-dimensional Euclidean spaces, with the usual metric are complete.
- Any compact metric space is sequentially compact and hence complete. The converse does not hold: for example, R is complete but not compact.
- In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete.
- The rational numbers Q are not complete. For example, the sequence (xn) defined by x0 = 1, xn+1 = 1 + 1/xn is Cauchy, but does not converge in Q.
Completion
Every metric space X has a completion which is a complete metric space in which X is isometrically embedded as a dense subspace. The completion has a universal property.
Examples
- The real numbers R are the completion of the rational numbers Q with respect to the usual metric of absolute distance.
Topologically complete space
Completeness is not a topological property: it is possible for a complete metric space to be homeomorphic to a metric space which is not complete. For example, the map
is a homeomorphism between the complete metric space R and the incomplete space which is the unit circle in the Euclidean plane with the point (0,-1) deleted. The latter space is not complete as the non-Cauchy sequence corresponding to t=n as n runs through the positive integers is mapped to a non-convergent Cauchy sequence on the circle.
We can define a topological space to be metrically topologically complete if it is homeomorphic to a complete metric space. A topological condition for this property is that the space be metrizable and an absolute Gδ, that is, a Gδ in every topological space in which it can be embedded.