The Banaschewski compactification of an approach space is of Wallman-Shanin-type

Abstract

For a topological zero-dimensional Hausdorff space (X, ) it is well known that the Banaschewski compactification ζ(X, ) is of Wallman-Shanin-type, meaning that there exists a closed basis (the collection of all clopen sets), such that the Wallman-Shanin compactification with respect to this closed basis is isomorphic to ζ(X, ). For an approach space (X, ) the Wallman-Shanin compactification W (X, ) with respect to a Wallman-Shanin basis (a particular basis of the lower regular function frame ) was introduced by R. Lowen and the second author. Recently, various constructions of the Banaschewski compactification known for a topological space were generalised to the approach case. Given a Hausdorff zero-dimensional approach space (X, ), constructions of the Banaschewski compactification ζ ∗(X, ) were developed by the authors. In this paper we construct a particular Wallman-Shanin basis for (X, ) and show that the Wallman-Shanin compactification with respect to this particular basis is isomorphic to ζ ∗(X, ).