Zero dimensionality of the Čech-Stone compactification of an approach space

Abstract

Given a Hausdorff zero-dimensional approach space X with gauge G, we investigate its Čech-Stone compactification β⁎(X) and characterise zero-dimensionality of the compactification. This is done by comparing β⁎(X) to the Banaschewski compactification ζ⁎(X). The notion of strongly zero-dimensional is introduced. For a Hausdorff zero-dimensional approach space X this property is shown to be equivalent to β⁎(X)=ζ⁎(X).When strongly zero-dimensional is combined with approach normal, then this yields a property which we call dM-approach normality. Both strongly zero-dimensional and approach normal are implied by dM-approach normal. For topological approach spaces we recover the well known relations between ultra normal, normal and strongly zero-dimensional. A zero-dimensional metric approach space is ultrametric and even the strong property, dM-approach normality, is fulfilled by any ultrametric space.