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 *.

**[1]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[2]**M. N. Bleicher and E. Marczewski,*Remarks on dependence relations and closure operators*, Colloq. Math.**9**(1962), 209–212. MR**0142489**, https://doi.org/10.4064/cm-9-2-209-212**[3]**M. N. Bleicher and G. B. Preston,*Abstract linear dependence relations*, Publ. Math. Debrecen**8**(1961), 55–63. MR**0130258****[4]**Japheth Hall Jr.,*The independence of certain axioms of structures in sets*, Proc. Amer. Math. Soc.**31**(1972), 317–325. MR**0291047**, https://doi.org/10.1090/S0002-9939-1972-0291047-X**[5]**J. Nagata,*Modern dimension theory*, Bibliotheca Math., vol. 6, Interscience, New York, 1965. MR**34**#8380.

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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