On permutable pairs of quasi-uniformities

Abstract

As it is well known, the concepts of normality and extremal disconnectedness of a topological space are dual to each other in some sense. This is nicely illustrated by several pairs of famous results in classical topology. A recent paper by E.P. de Jager and H.-P.A. Künzi provides some interesting pairs of results of the kind in the asymmetric setting of quasi-uniform spaces. The aim of this paper is to shed a more unifying light on these results. Besides extending them to a setting determined by more general fixed classes of subspaces of the underlying space, encompassing some weak variants of normality, we determine sufficient conditions on the fixed class of subspaces that enable us to unify each pair of results under the same proof.