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Topological spaces with a $ \sigma $-point finite base


Author: C. E. Aull
Journal: Proc. Amer. Math. Soc. 29 (1971), 411-416
MSC: Primary 54.50
DOI: https://doi.org/10.1090/S0002-9939-1971-0281154-9
MathSciNet review: 0281154
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Abstract: The principal results of the paper are as follows. A topological space with a $ \sigma $-point finite base has a $ \sigma $-disjoint base if it is either hereditarily collectionwise normal or hereditarily screenable. From a metrization theorem of Arhangel'skiĭ, it follows that a $ {T_1}$-space with a $ \sigma $-point finite base is metrizable iff it is perfectly normal and collectionwise normal. A topological space with a $ \sigma $-point base is quasi-developable in the sense of Bennett. Consequently a theorem of Čoban follows that for a topological space $ (X,\Im )$ the following are equivalent: (a) $ (X,\Im )$ is a metacompact normal Moore space, (b) $ (X,\Im )$ is a perfectly normal $ {T_1}$-space with a $ \sigma $-point finite base.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0281154-9
Keywords: $ \sigma $-point finite base, $ \sigma $-disjoint base, quasi-development, Moore space, hereditarily collectionwise normal, hereditarily screenable, metacompact or pointwise paracompact, subparacompact or $ {F_\sigma }$-screenable
Article copyright: © Copyright 1971 American Mathematical Society

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