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The Banaschewski compactification revisited

Tijdschriftbijdrage - Tijdschriftartikel

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.

Tijdschrift: J. Pure Appl. Algebra
ISSN: 0022-4049
Issue: 12
Volume: 223
Pagina's: 5185-5214
Jaar van publicatie:2019
CSS-citation score:1
Toegankelijkheid:Closed