Covering dimension in finitedimensional metric spaces
Author:
Japheth Hall
Journal:
Proc. Amer. Math. Soc. 41 (1973), 274277
MSC:
Primary 54F45
MathSciNet review:
0322828
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Abstract: Let be a structure in a topological space such that if , and is closed if . If is a covering of , let . If is a set and is a set, let denote the cardinal number of and . Let be an integer such that is defined as follows: if . If , then if (1) for each finite open covering of , there is an open refinement of such that if ; and (2) there is a finite open covering of such that if is an open refinement of , then for some . We say that has property () if for each nonempty open and each such that and whenever and each . THEOREM. If is a metric space, has property (), is finite, and if , then .
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 M. N. Bleicherand E. Marczewski, Remarks on dependence relations and closure operators, Colloq. Math. 9 (1962), 209212. MR 26 #58. MR 0142489 (26:58)
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 J. Hall, Jr., The independence of certain axioms of structures in sets, Proc. Amer. Math. Soc. 31 (1972), 317325. MR 45 #141. MR 0291047 (45:141)
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 J. Nagata, Modern dimension theory, Bibliotheca Math., vol. 6, Interscience, New York, 1965. MR 34 #8380.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197303228282
PII:
S 00029939(1973)03228282
Keywords:
Covering dimension,
covering dimension relative to structures in topological spaces,
finitedimensional metric spaces
Article copyright:
© Copyright 1973 American Mathematical Society
