Nonstandard methods of completing quasi-uniform spaces

Abstract

A rather general method of constructing T 2-completions of a quasi-uniform space is given. Using it we show that a sufficient condition for a quasi-uniform space to have a T 2-completion is that the quasi-uniformity contains a compatible uniformity. A necessary condition is also found: if F is a Cauchy filter, U ϵ V and x ϵ X, then there are F ϵ F,V,W ϵ V,V symmetric, such that {ifz ϵ F : ∃y ϵ W[F] such that ( x, y), ( y, z) ϵ V ⊂ U[x]. For locally compact spaces the method can be modified in order to obtain quasi-uniform T 2-compactifications.