Cardinalities of first countable $R$-closed spaces

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.