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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On ultrametrization of general metric spaces
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by Alex J. Lemin PDF
Proc. Amer. Math. Soc. 131 (2003), 979-989 Request permission


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:
  • 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
  • © Copyright 2002 American Mathematical Society
  • 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:
  • MathSciNet review: 1937437