14:30
The talk will focus on five items:
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.
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?
[`$\omega_1$-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 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 by "space" I mean "Hausdorff space" and so regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.
Theorem 2. The axiom $\square_{\aleph_1}$ implies that there is a normal, locally countable, countably compact space of cardinality $\aleph_2$.
This may be the first application of $\square_{\aleph_1}$ to construct a topological space whose existence in ZFC is unknown.