Which connected metric spaces are compact?
Author:
Gerald Beer
Journal:
Proc. Amer. Math. Soc. 83 (1981), 807811
MSC:
Primary 54E45; Secondary 54D05
MathSciNet review:
630059
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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.
 [1]
James
Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass.,
1966. MR
0193606 (33 #1824)
 [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 (24 #A2374)
 [3]
F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264286.
 [4]
George
F. Simmons, Introduction to topology and modern analysis,
McGrawHill Book Co., Inc., New YorkSan Francisco, Calif.TorontoLondon,
1963. MR
0146625 (26 #4145)
 [1]
 J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
 [2]
 M. H. A. Newman, Elements of topology of plane sets of points, Cambridge Univ. Press, New York, 1961. MR 0132534 (24:A2374)
 [3]
 F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264286.
 [4]
 G. Simmons, Introduction to topology and modern analysis, McGrawHill, New York, 1963. MR 0146625 (26:4145)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198106300596
PII:
S 00029939(1981)06300596
Article copyright:
© Copyright 1981
American Mathematical Society
