ABSTRACT: A space of countable extent, also called an omega_1-compact space, is one in which every closed discrete subspace is countable. The axiom used in the following theorem is consistent if it is consistent that there is a supercompact cardinal.
Theorem 1 The LCT axiom implies that every hereditarily normal, omega_1-compact space
is sigma-countably compact, i.e., the union of countably many countably compact subspaces.
Even for the specialized subclass of monotonically normal spaces, this is only a consistency result:
Theorem 2 If club, then there exists a locally compact, omega_1-compact monotonically
normal space that is not sigma-countably compact.
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.
The following unsolved problem is also discussed:
Problem Is there a ZFC example of a locally compact, omega_1-compact space
of cardinality aleph_1 that is not sigma-countably compact?
- Analytic Topology in Mathematics and Computer Science