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Proceedings of the American Mathematical Society

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Spaces in which compact sets have countable local bases


Author: Ralph R. Sabella
Journal: Proc. Amer. Math. Soc. 48 (1975), 499-504
MSC: Primary 54E35
DOI: https://doi.org/10.1090/S0002-9939-1975-0370522-6
MathSciNet review: 0370522
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Abstract: $ {D_1}$-spaces and coconvergent spaces are examples of spaces in which compact sets have countable local bases ($ {D_0}$-spaces). Among the results related to $ {D_0}$-spaces given in this paper is a sufficient condition under which such spaces are coconvergent. In relation to a question posed by F. B. Jones, it is shown that a topological property sufficient for semimetrizable spaces to be developable is that they be coconvergent. Coconvergence implies metrizability in stratifiable spaces; it is shown in this paper that a $ {D_0}$-space is metrizable if there exists a stratification satisfying a nesting-like condition.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0370522-6
Keywords: Open neighborhood assignments, $ U$-linked sequences, coconvergent, contraconvergent, stratifiable spaces, Nagata spaces
Article copyright: © Copyright 1975 American Mathematical Society

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