On ultrametrization of general metric spaces

Abstract:

This paper gives a complete description of ultrametric spaces up to uniform equivalence. It also describes all metric spaces which can be mapped onto ultrametric spaces by a non-expanding one-to-one map. Moreover, it describes particular classes of spaces, for which such a map has a continuous (uniformly continuous) inverse map. This gives a complete solution for the Hausdorff-Bayod Problem (what metric spaces admit a subdominant ultrametric?) as well as for two other problems posed by Bayod and Martínez-Maurica in 1990. Further, we prove that for any metric space $(X,d)$, there exists the greatest non-expanding ultrametric image of $X$ (an ultrametrization of $X$), i.e., the category of ultrametric spaces and non-expanding maps is a reflective subcategory in the category of all metric spaces and the same maps. In Section II, for any cardinal $\tau$, we define a complete ultrametric space $L_\tau$ of weight $\tau$ such that any metric space $X$ of weight $\tau$ is an image of a subset $L(X)$ of $L_\tau$ under a non-expanding, open, and compact map with totally-bounded pre-images of compact subsets. This strengthens Hausdorff-Morita, Morita-de Groot and Nagami theorems. We also construct an ultrametric space $L(\tau )$, which is a universal pre-image of all metric spaces of weight $\tau$ under non-expanding open maps. We define a functor $\lambda$ from the category of ultrametric spaces to a category of Boolean algebras such that algebras $\lambda (X)$ and $\lambda (Y)$ are isomorphic iff the completions of spaces $X$ and $Y$ are uniformly homeomorphic. Some properties of the functor $\lambda$ and the ultrametrization functor are discussed.