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?

18 May 2016

16:00

Peter Nyikos

Abstract