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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0630059-6
PII: S 0002-9939(1981)0630059-6
Article copyright: © Copyright 1981 American Mathematical Society