Locally Compact, Ω1-compact Spaces

This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo large cardinals) that a locally compact space X must be the union of countably many ω-bounded subspaces if every closed discrete subspace of X is countable [in other words, if X is ω1-compact]? A space is ω-bounded if every countable subset has compact closure. This is a strengthening of countable compactness. Assuming the consistency of a supercompact cardinal, it is shown that the answer is affirmative for the specialized class of monotonically normal spaces and the more general class of hereditarily ω1-scwH spaces, where “scwH” stands for “strongly collectionwise Hausdorff”. Even for monotonically normal spaces, this is only a consistency result: Theorem. If ♣, then there exists a locally compact, ω1-compact monotonically normal space that is not the union of countably many ω-bounded subspaces. These two results together are unusual in that most independence results on monotonically normal spaces depend on whether Souslin’s Hypothesis (SH) is true, and do not involve large cardinal axioms. Here, it is not known whether either SH or its negation affect either direction in this independence result. It is shown that if the answer to the above problem is Yes, then the cardinal c of the continuum must be at least א3. But it is not known whether this is still true if X is normal in addition to being locally compact. Introduction. In [EN], we introduced the following powerful consequence of what has since come to be known as the P-Ideal Dichotomy (PID) axiom, and showed many of its topological implications, one of which will be recalled in the Appendix: The Locally Compact Trichotomy (LCT) axiom. Every locally compact space has either: 1991 Mathematics Subject Classification. (Updated to 2014) Primary: 03E35, 54D15, 54D45, 54D99 Secondary: 03E55, 54C99, 54D20.