Entourages, Density, Cauchy Maps, and Completion

Abstract

We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part.

In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one).

Keywords

• Frame
• Locale
• Sublocale
• Uniform frame
• Quasi-uniform frame
• Entourage
• Uniform map
• Uniform dense embedding
• Cauchy map
• Complete
• Completion