In what follows, "space" means "Hausdorff (\(T_2\)) topological space."

Some of the theorems and problems to be discussed include:

Theorem 1. It is ZFC-independent whether every locally compact, \(\omega_1\)-compact space of cardinality \(\aleph_1\) is the union of countably many countably compact spaces.

[‘\(\omega_1\)-compact’ means that every closed discrete subspace is countable. This is obviously implied by being the union of countably many countably compact spaces, but the converse is not true.]

Problem 1. Is it consistent that every locally compact, \(\omega_1\)-compact space of cardinality \(\aleph_2\) is the union of countably many countably compact spaces? Problem 2. Is ZFC enough to imply that there is a normal, locally countable, countably compact space of cardinality greater than \(\aleph_1\)?

Problem 3. Is it consistent that there exists a normal, locally countable, countably compact space of cardinality greater than \(\aleph_2\)?

The spaces involved in Problem 2 and Problem 3 are automatically locally compact, because regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.

Problem 4 [Problem 5]. Is there an upper bound on the cardinalities of regular [resp. normal], locally countable, countably compact spaces?

Theorem 2. The axiom \(\square_\) implies that there is a normal, locally countable, countably compact space of cardinality \(\aleph_2\).

The statement in Theorem 1 was shown consistent by Lyubomyr Zdomskyy, assuming \(\mathfrak p > \aleph_1\) plus P-Ideal Dichotomy (PID). Counterexamples have long been known to exist under \(\mathfrak b = \aleph_1\), under \(\clubsuit\), and under the existence of a Souslin tree.

Theorem 2 may be the first application of \(\square_\) to construct a topological space whose existence in ZFC is unknown.