complete uniform space CompleteUniformSpace 2007-02-12 16:49:41 2007-03-10 00:19:03 Definition Cauchy filter Cauchy sequence sequentially complete complete uniformity \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here \PMlinkescapeword{complete} Let $X$ be a uniform space with uniformity $\mathcal{U}$. A filter $\mathcal{F}$ on $X$ is said to be a \emph{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 \emph{complete} if every Cauchy filter is a convergent filter in the topology $T_{\mathcal{U}}$ \PMlinkname{induced}{TopologyInducedByUniformStructure} by $\mathcal{U}$. $\mathcal{U}$ in this case is called a \emph{complete uniformity}. A \emph{Cauchy sequence} $\lbrace x_i\rbrace$ 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 \emph{sequentially complete} if every Cauchy sequence converges (every section filter of it converges). \textbf{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.