Which connected metric spaces are compact?

Author:
Gerald Beer

Journal:
Proc. Amer. Math. Soc. **83** (1981), 807-811

MSC:
Primary 54E45; Secondary 54D05

DOI:
https://doi.org/10.1090/S0002-9939-1981-0630059-6

MathSciNet review:
630059

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Abstract | References | Similar Articles | Additional Information

Abstract: A metric space is called chainable if for each each two points in can be joined -chain. is called uniformly chainable if for there exists an integer such that each two points can be joined -chain of length at most .

Theorem. *A chainable metric space* *is a continuum if and only if* *is uniformly chainable and there exists* *such that each closed* -ball is compact.

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

**[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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DOI:
https://doi.org/10.1090/S0002-9939-1981-0630059-6

Article copyright:
© Copyright 1981
American Mathematical Society