Which connected metric spaces are compact?
Proc. Amer. Math. Soc. 83 (1981), 807-811
Primary 54E45; Secondary 54D05
Full-text PDF Free Access
Similar Articles |
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.
Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass.,
0193606 (33 #1824)
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)
F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286.
F. Simmons, Introduction to topology and modern analysis,
McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London,
0146625 (26 #4145)
- J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
- M. H. A. Newman, Elements of topology of plane sets of points, Cambridge Univ. Press, New York, 1961. MR 0132534 (24:A2374)
- F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286.
- G. Simmons, Introduction to topology and modern analysis, McGraw-Hill, New York, 1963. MR 0146625 (26:4145)
Retrieve articles in Proceedings of the American Mathematical Society
Retrieve articles in all journals
© Copyright 1981
American Mathematical Society