Two Ä-closed Spaces Revisited

Recently, R. M. Stephenson has used the Continuum Hypothesis to construct two Ä-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, it is at least consistent with the negation of the Continuum Hypothesis that there exist spaces with the same properties, but which have cardinality K,. We conclude with some consistency results concerning relationships between open filter bases and generalizations of the notions of feeble compactness and Lindelöfness. In [8, §4] R. M. Stephenson constructs two spaces (S,§>) and (T,?T) which are both separable, first countable, feebly compact, not countably compact, regular, and Hausdorff. Further, (S,i>) is not minimal regular, but if the Continuum Hypothesis holds, then (S, S) is .R-closed and (T, 5") is strongly minimal regular. We shall prove that the construction may be modified in such a way as to insure that the spaces retain these latter properties without the assumption of any special set theoretical hypotheses (other than the Axiom of Choice which we shall assume throughout and without further mention), and we shall prove that it is consistent with the negation of the Continuum Hypothesis that these spaces have cardinality N,. These spaces are of importance because they appear to be the only known examples of separable first countable Ä-closed spaces which are not compact. We shall also consider consistency results concerning generalizations of the notions of feeble compactness and Lindelöfness. These will give us, for example, conditions under which every open filter over a space has an adherent point. Although many of our results will be topological in nature, our constructions will be set theoretical and will be self-contained. However, because Stephenson's original construction, while very clever, is quite involved, we shall not repeat it here, and a detailed knowledge of [8] would be required to reconstruct the two Ä-closed spaces referred to in the title. In what follows, all spaces are assumed to be at least Tx. In Stephenson's construction, the Continuum Hypothesis is used to couple two lemmas. One (Lemma 4.1) states that in a certain space there is a subset Received by the editors November 18, 1974 and, in revised form, March 3, 1975 and August 7, 1975. AM S (MOS) subject classifications (1970). Primary 54A25, 54D25; Secondary 02K25, 04A30. 1 The preparation of this paper was supported by a grant from the City University of New York Faculty Research Program. © American Mathematical Society 1976 303 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use