complete uniform space

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

We say that $X$ is complete if every Cauchy filter is a convergent filter in the topology $T_{𝒰}$ induced (http://planetmath.org/TopologyInducedByUniformStructure) by $𝒰$. $𝒰$ in this case is called a complete uniformity.

A Cauchy sequence $\{x_i\}$ 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 converges (every section filter of it converges).

Remark. This is a generalization 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