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Maps on D1 and D2 spaces

Part of: Basic constructions Generalities

Published online by Cambridge University Press:  09 April 2009

H. B. Potoczny
H. B. Potoczny
Affiliation:
Wright-Patterson AFB Ohio 45433, U.S.A.
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Abstract

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A space X is said to be D1 provided each closed set has a countable basis for the open sets containing it. It is said to be D2 provided there is a countable base {Un} such that each closed set has a countable base for the open sets containing it, which is a subfamily of {Un}. In this paper, we give a separation theorem for D1 spaces, and provide a characterization of D1 and D2 spaces in terms of maps.

MSC classification

Secondary: 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets) 54B15: Quotient spaces, decompositions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 1 , August 1984 , pp. 106 - 109
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Aull, C. E., ‘Closed set countability axioms’, Indag. Math. 28 (1966), 311–316.CrossRefGoogle Scholar
[2]Aull, C. E., ‘Compactness as a base axiom’, Indag. Math. 29 (1967), 106–108.CrossRefGoogle Scholar
[3]Warrack, B. and Willard, S., ‘Domains of first countability’, Glasnik Mat. Ser. III 14 (36) (1979), 129–139.Google Scholar