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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Two $R$-closed spaces revisited
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by Stephen H. Hechler PDF
Proc. Amer. Math. Soc. 56 (1976), 303-309 Request permission

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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 303-309
  • MSC: Primary 54D25; Secondary 54A25
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0405354-4
  • MathSciNet review: 0405354