On ultrametrization of general metric spaces

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 ) (X,d) , there exists the greatest non-expanding ultrametric image of X X ( an ultrametrization of X 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 τ \tau , we define a complete ultrametric space L τ L_\tau of weight τ \tau such that any metric space X X of weight τ \tau is an image of a subset L ( X ) L(X) of L τ 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 ( τ ) L(\tau ) , which is a universal pre-image of all metric spaces of weight τ \tau under non-expanding open maps. We define a functor λ \lambda from the category of ultrametric spaces to a category of Boolean algebras such that algebras λ ( X ) \lambda (X) and λ ( Y ) \lambda (Y) are isomorphic iff the completions of spaces X X and Y Y are uniformly homeomorphic. Some properties of the functor λ \lambda and the ultrametrization functor are discussed.