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Proceedings of the American Mathematical Society

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



On maximal groups of isometries

Author: Ludvík Janoš
Journal: Proc. Amer. Math. Soc. 28 (1971), 584-586
MSC: Primary 54.80
MathSciNet review: 0275403
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Abstract: The purpose of this note is to introduce the concept of ``Optimal Metrization'' for metrizable topological spaces. Let $ X$ be such a space, $ \rho $ a metric on $ X$ and $ K(\rho )$ the group of all those homeomorphisms of $ X$ onto itself which preserve $ \rho $. The metric $ \rho $ is said to be ``optimal'' provided there is no $ {\rho ^ \ast }$ with $ K({\rho ^ \ast })$ properly containing $ K(\rho )$. A space having at least one optimal metric is called ``optimally metrizable.'' Examples of spaces which are and which are not optimally metrizable are given; it is shown that the real line $ R$ is, and that the usual metric is optimal.

References [Enhancements On Off] (What's this?)

  • [1] S. Eilenberg, Sur les groupes compacts d'homéomorphies, Fund. Math. 28 (1937), 75-80.
  • [2] Martin T. Wechsler, Homeomorphism groups of certain topological spaces, Ann. of Math. (2) 62 (1955), 360–373. MR 0072453

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Keywords: Group of isometries, optimal metric, optimally metrizable
Article copyright: © Copyright 1971 American Mathematical Society