Locally compact, $\omega_1$-compact spaces

Abstract

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$ is $\sigma$-countably compact. Whether $\aleph_1$ can be replaced with $\aleph_2$ is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, $\omega_1$-compact space is $\sigma$-countably compact. As a result, it is also ZFC-independent whether there is a locally compact, $\omega_1$-compact Dowker space of cardinality $\aleph_1$, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space $\omega_1$. Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as ``MM(S)[S]''. Most of the work is one by the $P$-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality $\aleph_1$, as it is in several theorems.