Uniform Space - Completeness

Completeness

Generalising the notion of complete metric space, one can also define completeness for uniform spaces. Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets).

A Cauchy filter F on a uniform space X is a filter F such that for every entourage U, there exists A∈F with A×A ⊆ 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 Cauchy filter is called minimal if it contains no smaller (i.e., 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.

Conversely, a uniform space is called complete if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.

Complete uniform space enjoy the following important property: if f: A → Y is a uniformly continuous function from a dense subset A of a uniform space X into a complete uniform space Y, then f can be extended (uniquely) into a uniformly continuous function on all of X.

A topological space which can be made into a complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space.