The Banaschewski compactification revisited

Abstract

Given an ultrametric space (X,d) and its related distance δd between points and subsets of X, we construct an extension (Z,δζ), where Z is the underlying set of the Smirnov compactification of a suitable proximity on X and δζ is a distance between points and subsets of Z. The underlying topological space defined by the closure p∈clζA if and only if δζ(p,A)=0, for a point p∈Z and a subset A⊆Z, is a Hausdorff zero-dimensional compactification of the underlying topological space of (X,d) and the distance δζ is an extension of the distance δd. The construction is performed in the larger category ZDApp of Hausdorff zero-dimensional approach spaces with as objects sets structured with a distance, subject to suitable axioms and with contractions as morphisms. Both the categories UMet of ultrametric spaces with non-expansive maps and ZDim of zero-dimensional topological spaces and continuous maps are fully embedded in ZDApp. In this broader setting we obtain an approach counterpart for the Banaschewski compactification which is a reflection ZDApp2→kZDApp2, from Hausdorff zero-dimensional approach spaces to Hausdorff compact zero-dimensional approach spaces. In the topological case the construction coincides with the topological Banaschewski compactification. In the ultrametric case the space (Z,δζ) retains numerical information from (X,d), whereas the topological Banaschewski compactification applied to the underlying topological space of (X,d) is generally not (ultra) metrisable.