The Banaschewski compactification revisited

Given an ultrametric space (X,d) and its related distance δ _d between points and subsets of X, we construct an extension (Z,δ ^ζ), where Z is the underlying set of the Smirnov compactification of a suitable proximity on X and δ ^ζ is a distance between points and subsets of Z. The underlying topological space defined by the closure p∈cl ^ζA if and only if δ ^ζ(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 δ ^ζ is an extension of the distance δ _d. 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 ZDApp ₂→kZDApp ₂, 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,δ ^ζ) 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.