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,
).