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Proceedings of the American Mathematical Society

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



Two $R$-closed spaces revisited

Author: Stephen H. Hechler
Journal: Proc. Amer. Math. Soc. 56 (1976), 303-309
MSC: Primary 54D25; Secondary 54A25
MathSciNet review: 0405354
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Abstract: Recently, R. M. Stephenson has used the Continuum Hypothesis to construct two $R$-closed, separable regular, first countable, noncompact Hausdorff spaces. We show that the assumption of the Continuum Hypothesis can be removed by replacing a lemma used in the original construction to deal with arbitrary almost-disjoint families by the construction of a particular almost-disjoint family. We also show that while these spaces always have cardinality ${\mathbf {c}}$, it is at least consistent with the negation of the Continuum Hypothesis that there exist spaces with the same properties, but which have cardinality ${\aleph _1}$. We conclude with some consistency results concerning relationships between open filter bases and generalizations of the notions of feeble compactness and Lindelöfness.

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Article copyright: © Copyright 1976 American Mathematical Society