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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The $ Q$-topology, Whyburn type filters and the cluster set map


Author: Robert A. Herrmann
Journal: Proc. Amer. Math. Soc. 59 (1976), 161-166
MSC: Primary 54J05
DOI: https://doi.org/10.1090/S0002-9939-1976-0482730-5
Erratum: Proc. Amer. Math. Soc. 65 (1977), 375.
MathSciNet review: 0482730
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Abstract: We use nonstandard topology and the $ Q$-topology to characterize normal, almost-normal, regular, almost-regular, semiregular spaces. The cluster [resp. $ \theta $-cluster] set relation is used to characterize regular, almost-regular [resp. strongly-regular] spaces. The Whyburn [resp. Dickman] filter bases are characterized and it is shown that the cluster [resp. $ \theta $-cluster] set relation restricted to the domain of the Whyburn [resp. Dickman] filter bases is an essentially continuous [resp. strongly $ \theta $-continuous] map iff the space is Hausdorff [resp. Urysohn].


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DOI: https://doi.org/10.1090/S0002-9939-1976-0482730-5
Keywords: $ Q$-topology, Whyburn filters, Dickman filters, regular, almost-regular, strongly-regular, semiregular, normal, almost-normal, cluster set map, $ \theta $-cluster set map, directed toward, almost-convergent to
Article copyright: © Copyright 1976 American Mathematical Society