The Banaschewski compactification is of Wallman-Shanin type

For a topological zero-dimensional Hausdorff space (X, (Figure presented.)) it is well known that the Banaschewski compactification ζ(X, (Figure presented.)) 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, (Figure presented.)). For an approach space (X, (Figure presented.)) the Wallman-Shanin compactification W (X, (Figure presented.)) with respect to a Wallman-Shanin basis (Figure presented.) (a particular basis of the lower regular function frame (Figure presented.)) 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, (Figure presented.)), constructions of the Banaschewski compactification ζ ^∗(X, (Figure presented.)) were developed by the authors. In this paper we construct a particular Wallman-Shanin basis for (X, (Figure presented.)) and show that the Wallman-Shanin compactification with respect to this particular basis is isomorphic to ζ ^∗(X, (Figure presented.)).