A quasi-uniform characterization of Wallman type compactifications

Abstract

A *-compactification of aT1quasi-uniform space (X,U) is a compactT1quasi-uniform space (Y,V) that has aT(V*)-dense subspace quasi-isomorphic to (X,U), whereV* denotes the coarsest uniformity finer thanV.In this paper we characterize all Wallman type compactifications of aT1topological space in terms of the *-compactification of its point symmetric totally bounded transitive compatible quasi-uniformities. We deduce that the *-compactification of the Pervin quasi-uniformity of any normalT1topological spaceXis exactly the Stone-Cech compactification ofX. We also obtain a characterization of those Hausdorff compactifications of a given space, which are of Wallman type.