Dimension zero at all scales

Abstract

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.