Covering dimension in finite-dimensional metric spaces

Author:
Japheth Hall

Journal:
Proc. Amer. Math. Soc. **41** (1973), 274-277

MSC:
Primary 54F45

DOI:
https://doi.org/10.1090/S0002-9939-1973-0322828-2

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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0322828-2

Keywords:
Covering dimension,
covering dimension relative to structures in topological spaces,
finite-dimensional metric spaces

Article copyright:
© Copyright 1973
American Mathematical Society