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
Fulltext PDF Free Access
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 (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 York, 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
54E45,
54D05
Retrieve articles in all journals
with MSC:
54E45,
54D05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198106300596
PII:
S 00029939(1981)06300596
Article copyright:
© Copyright 1981 American Mathematical Society
