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On ultrametrization of general metric spaces

Author: Alex J. Lemin
Journal: Proc. Amer. Math. Soc. 131 (2003), 979-989
MSC (2000): Primary 54E35, 54E05, 54E40, 54E50; Secondary 06B30, 06E15, 11E95, 12J25, 18A40, 18B30, 26E30, 54B30, 54C10, 54D30
Published electronically: October 18, 2002
MathSciNet review: 1937437
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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.

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Additional Information

Alex J. Lemin
Affiliation: Department of Mathematics, Moscow State University of Civil Engineering, 26 Yaro- slavskoe Highway, Moscow 129337, Russia

Keywords: Metric space, ultrametric space, proximity space, complete space, compactification, Smirnov compactification, uniform equivalence, non-expanding map, subdominant ultrametric, ultrametrization of metric space, Boolean algebra, category, functor, reflective functor, ultrametrization functor, isomorphism of categories
Received by editor(s): December 30, 2000
Received by editor(s) in revised form: October 29, 2001
Published electronically: October 18, 2002
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society

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