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

DOI:
https://doi.org/10.1090/S0002-9939-02-06605-4

Published electronically:
October 18, 2002

MathSciNet review:
1937437

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Email:
alex_lemin@hotmail.com

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