Baire resolvability on Hausdorff spaces

David Fernández-Duque, Ghent University. Part of the Logic Seminar series.

Joint work with Guram Bezhanishvili

A topological space is resolvable if it can be partitioned into two dense sets.  Hewitt showed that if a crowded space is either metric or locally compact Hausdorff, then it is resolvable.  One may then ask if the two sets may be chosen so that they are not only dense, but everywhere "large" in some sense; specifically, we consider largeness in the sense of Baire.

Recall that a set is meager if it can be written as a countable union of nowhere-dense sets.  Let us say that a set is "Baire-dense" if its intersection with any non-empty open set is non-meager, and a topological space is Baire-resolvable if it can be partitioned into two Baire-dense sets.  We show that, under some natural conditions, Hewitt's result can be strengthened to obtain Baire-resolvability of crowded spaces. Finally, we discuss how this result can be seen within the more general context of Kuratowski algebras, which are Boolean algebras equipped with a closure-like operation.