# complete uniform space

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

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