Which connected metric spaces are compact?
Proc. Amer. Math. Soc. 83 (1981), 807-811
Primary 54E45; Secondary 54D05
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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.
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- M. H. A. Newman, Elements of topology of plane sets of points, Cambridge Univ. Press, New York, 1961. MR 0132534 (24:A2374)
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- G. Simmons, Introduction to topology and modern analysis, McGraw-Hill, New York, 1963. MR 0146625 (26:4145)
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