Sobrification and bicompletion of totally bounded quasi-uniform spaces

Abstract

Abstract We observe that if is a compatible totally bounded quasi-uniformity on a T 0 -space ( X , ), then the bicompletion of ( X , ) is a strongly sober, locally quasicompact space. It follows that the b -closure S of ( X , ) in is homeomorphic to the sobrification of the space ( X , ). We prove that S is equal to if and only if ( X , ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X , then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity on .