Regularity of Richardson's Compactification

Abstract

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.

A Compactification with θ-Continuous Lifting Property

“…A r 3 topological space X is defined to be a G-space if each non-convergent ultrafilter has a filter base of closed sets. This condition (but not the terminology) is due to Gazik [2], who showed that the pretopological compactification X* of a TYtopological space X is equivalent to fiX if and only if X is a G-space. When X* is a topological space, X* = X A .…”

“…When X* is a topological space, X* = X A . Thus if X is a G-space, it follows from [2] that X A is equivalent to I3X.…”