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.Comment: 21 pages, submitted, comments are welcom

Nyikos, Peter

Zdomskyy, Lyubomyr

English

arXiv

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.Comment: 20 pages, revised, comments are welcom

Locally compact, $\omega_1$ -compact spaces

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