Two 𝑅-closed spaces revisited

Hechler, Stephen H.

doi:10.1090/S0002-9939-1976-0405354-4

Two $R$-closed spaces revisited
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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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Zur Theorie der trigonometrischen Reihen

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