A quasi-uniform characterization of Wallman type compactifications

Abstract

A *-compactification of a T1 quasi-uniform space (X,U) is a compact T1 quasi-uniform space (Y,V) that has a T(V*)-dense subspace quasi-isomorphic to (X,U), where V* denotes the coarsest uniformity finer than V.In this paper we characterize all Wallman type compactifications of a T1 topological 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 normal T1 topological space X is exactly the Stone-Cech compactification of X. We also obtain a characterization of those Hausdorff compactifications of a given space, which are of Wallman type.