Countable dense homogeneous semifilters

Speaker(s)

Lyubomyr Zdomskyy

Affiliation

Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien

Language of the talk

English

Date

Nov. 4, 2026, 4:15 p.m.

Seminar

Topology and Set Theory Seminar

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A topological space is called countable dense homogeneous (CDH) if any two countable dense subsets can be mapped onto each other by a homeomorphism of the space onto itself. More than a decade ago, Hernández-Gutiérrez and Hrušák proved that every non-meager P-filter is CDH, and with  Kunen and Medini we have subsequently established the converse. The situation for semifilters remains considerably less understood, despite several results concerning their topological homogeneity. In particular, it is known that every Borel CDH semifilter is Polish and hence comeager. In this talk, we present a ZFC construction of a CDH semifilter that is not comeager. Since every CDH semifilter is non-meager, our example is neither meager nor comeager. We discuss this result in the context of the known characterizations of CDH filters and the broader problem of characterizing CDH semifilters.

The talk is going to be based on a joint work in progress with A. Medini.

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