# Totally imperfect Menger sets

- Speaker(s)
**Piotr Szewczak**- Affiliation
- Cardinal Stefan Wyszyński University in Warsaw
- Date
- Jan. 11, 2023, 4:15 p.m.
- Room
- room 5050
- Seminar
- Topology and Set Theory Seminar

A set of reals X is Menger if for any countable sequence of open covers of X one can pick finitely many elements from every cover in the sequence such that the chosen sets cover X. Any set of reals of cardinality smaller than the dominating number d is Menger and there is a non-Menger set of cardinality d. By the result of Bartoszyński and Tsaban, in ZFC, there is a totally imperfect (with no copy of the Cantor set inside) Menger set of cardinality d. We solve a problem, whether there is such a set of cardinality continuum. Using an iterated Sacks forcing and topological games we prove that it is consistent with ZFC that d is less than c and each totally imperfect Meneger set has cardinality less or equal than d. This is a joint work with Lyubomyr Zdomskyy. The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122