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Strong realcompactness and weakly measurable cardinals. (English) Zbl 0698.54019

Summary: Denote \(\beta X-X\) by \(X^*\). Define properties \(P_ 0\) and \(P_ 1\) of a space X by \(P_ i\) \((i<2):\) if \(D\subseteq X^*\) is countable and relatively discrete, and if \(\bar D\) is compact, then (1) if \(i=0:\) \(\bar D\neq \alpha D\), the smallest (=one-point) compactification of D, (2) if \(i=1:\) \(\bar D=\beta D\), the biggest (=Čech-Stone) compactification of D. (Note: X has \(P_ 0\) iff \(X^*\) has no nontrivial convergent sequences.)
Realcompact spaces are known to have \(P_ 0\), and a dense-in-itself metrizable space with \(P_ 0\) is realcompact. There is a separable first countable locally compact realcompact normal space without \(P_ 1\). The question of whether there is a realcompact metrizable space without \(P_ 1\) leads to a new small large cardinal. There is one iff there is a weakly measurable cardinal that is not measurable; a cardinal \(\kappa\) is called weakly measurable if there is a countable set \({\mathcal F}\) of free ultrafilters on \(\kappa\) such that the filter \(\cap {\mathcal F}\) is \(\kappa\)-complete. Let \(\kappa\) be the first weakly measurable cardinal that is not measurable (if there is one). We show that \(\kappa <{\mathfrak c}\) and \(\kappa\neq cf {\mathfrak c}\) and that \(Con(ZFC+\exists\) measurable)\(\to Con(ZFC+\kappa\) exists).

MSC:

54D60 Realcompactness and realcompactification
03E35 Consistency and independence results
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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