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Pair-Countable, Closure-Preserving Covers of Compact Sets

Part of: Fairly general properties

Published online by Cambridge University Press:  09 April 2009

Henry Potoczny
Henry Potoczny
Affiliation:
Department of Mathematics and Computer Science U.S. Air Force Institute of Technology Wright-Patterson, Ohio 45433
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Abstract

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In this paper, we prove the following results: (1) if a topological space X has a pair-countable, closure-preserving cover of compact sets, then X is locally paracompact at each point of X and X has a dense open subspace which is locally σ-compact. In addition, if X is also collectionwise-T2, then X is paracompact. Locally paracompact is taken to mean that each point X has an open set with paracompact closure.

MSC classification

Secondary: 54D45: Local compactness, $sigma$-compactness
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 1 , February 1988 , pp. 138 - 142
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Heath, R., ‘Screenability, pointwise paracompactness, and metrization of Moore spaces’, Canad. J. Math. 16 (1964), 763–770.CrossRefGoogle Scholar
[2]Katuta, Y., ‘On spaces which admit closure-preserving covers by compact sets’, Proc. Japan Acad. 50, No. 10 (1974).Google Scholar
[3]Lutzer, D., ‘Pixley-Roy topology’, Topology Proc. 3 (1978), 139–158.Google Scholar
[4]Potoczny, H., ‘A nonparacompact space which admits a closure-preserving cover of compact sets’, Proc. Amer. Math. Soc. 32 (1972), 309–311.CrossRefGoogle Scholar
[5]Potoczny, H., ‘On a problem of Tamano’, Fund. Math. 75 (1972), 29–31.CrossRefGoogle Scholar
[6]Potoczny, H., Closure-preserving families of compact sets, General Topology Appl. 3 (1973), 243–248.CrossRefGoogle Scholar
[7]Potoczny, H. and Junnila, H., ‘Closure-preserving families and metacompactness’, Proc. Amer. Math. Soc. 53 (1975), 523–529.CrossRefGoogle Scholar
[8]Tamano, H., ‘A characterization of paracompactness’, Fund. Math. 72 (1971), 189–201.CrossRefGoogle Scholar