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Which connected metric spaces are compact?

Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 83 (1981), 807-811
MSC: Primary 54E45; Secondary 54D05
MathSciNet review: 630059
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Abstract: A metric space $ X$ is called chainable if for each $ \varepsilon > 0$ each two points in $ X$ can be joined $ \varepsilon $-chain. $ X$ is called uniformly chainable if for $ \varepsilon $ there exists an integer $ n$ such that each two points can be joined $ \varepsilon $-chain of length at most $ n$.

Theorem. A chainable metric space $ X$ is a continuum if and only if $ X$ is uniformly chainable and there exists $ \delta > 0$ such that each closed $ \delta $-ball is compact.

Using Ramsey's Theorem a sequential characterization of uniformly chainable metric spaces is obtained, paralleling the one for totally bounded spaces.

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

  • [1] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
  • [2] M. H. A. Newman, Elements of the topology of plane sets of points, Second edition, reprinted, Cambridge University Press, New York, 1961. MR 0132534
  • [3] F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286.
  • [4] George F. Simmons, Introduction to topology and modern analysis, McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London, 1963. MR 0146625

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Article copyright: © Copyright 1981 American Mathematical Society

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