Abstract
Recently, R. M. Stephenson has used the Continuum Hypothesis to construct two R 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 c {\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 âľ 1 {\aleph _1} . We conclude with some consistency results concerning relationships between open filter bases and generalizations of the notions of feeble compactness and LindelĂśfness.
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