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Cardinalities of first countable $ R$-closed spaces

Authors: Alan Dow and Jack Porter
Journal: Proc. Amer. Math. Soc. 89 (1983), 527-532
MSC: Primary 54D25; Secondary 54A25
MathSciNet review: 715880
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Abstract: It is now well known that first countable compact Hausdorff spaces are either countable or have cardinality $ c$. The situation for first countable $ H$-closed spaces is that they have cardinality less than or equal $ c$, and it is at least consistent that they may have cardinality $ {\aleph _1} < c$. We show that the situation is quite different for first countable $ R$-closed spaces. We begin by constructing an example which has cardinality $ {\aleph _1}$. Let $ {\lambda _0}$ be the smallest cardinal greater than $ c$ which is not a successor. For each cardinal $ \kappa $ with $ c \leqslant \kappa \leqslant {\lambda _0}$ we construct a first countable $ R$-closed space of cardinality $ \kappa $. We also construct a first countable $ R$-closed space of cardinality $ \lambda _0^\omega $. This seems to indicate that there is no reasonable upper bound to the cardinalities of $ R$-closed spaces as a function of their character.

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Article copyright: © Copyright 1983 American Mathematical Society

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