complete uniform space

Let X be a uniform space with uniformity 𝒰. A filter on X is said to be a Cauchy filter if for each entourage V in 𝒰, there is an F such that F×FV.

We say that X is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if every Cauchy filter is a convergent filter in the topologyMathworldPlanetmathPlanetmath T𝒰 induced ( by 𝒰. 𝒰 in this case is called a complete uniformity.

A Cauchy sequence {xi} in a uniform space X is a sequence in X whose section filter is a Cauchy filter. A Cauchy sequence is said to be convergent if its section filter is convergent. X is said to be sequentially complete if every Cauchy sequence convergesPlanetmathPlanetmath (every section filter of it converges).

Remark. This is a generalizationPlanetmathPlanetmath of the concept of completeness in a metric space, as a metric space is a uniform space. As we see above, in the course of this generalization, two notions of completeness emerge: that of completeness and sequentially completeness. Clearly, completeness always imply sequentially completeness. In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.

Title complete uniform space
Canonical name CompleteUniformSpace
Date of creation 2013-03-22 16:41:40
Last modified on 2013-03-22 16:41:40
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 54E15
Synonym semicomplete
Synonym semi-complete
Related topic Complete
Defines Cauchy filter
Defines Cauchy sequence
Defines sequentially complete
Defines complete uniformity