Uniformizable Cauchy spaces

Abstract

A family C of filters on a set X is uniformizable if there is a uniformity on X such that C is its collection of Cauchy filters. Using the theory of completions and Cauchy continuous maps for Cauchy spaces, we obtain characterizations of uniformizable Cauchy spaces. In particular, given a Cauchy structure C on X we investigate under what conditions the filter u(C)=⋂F∈CF×F is a uniformity and C is its collection of Cauchy filters. This problem is treated using Cauchy covering systems.

Keywords

• uniformity
• Cauchy space
• completion
• Cauchy
• continuous map
• realcompactness
• Cauchy covering system.