Cauchy filter

Completeness

A Cauchy filter (respectively, a Cauchy prefilter) on a uniform space ${\displaystyle X}$ is a filter (respectively, a prefilter) ${\displaystyle F}$ such that for every entourage ${\displaystyle U,}$ there exists ${\displaystyle A\in F}$ with ${\displaystyle A\times A\subseteq U.}$ In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter. A minimal Cauchy filter is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique minimal Cauchy filter. The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.