Piotr Szewczak: Totally imperfect Menger sets
31/05/23 10:19
Tuesday, June 6, 2023 17:00
Location: room A.4.1 C-19
Speaker: Piotr Szewczak (UKSW)
Title: Totally imperfect Menger sets
Abstract: 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 \(d \lt c\) and each totally imperfect Menger set has cardinality less or equal than \(d\).
This is a joint work with Valentin Haberl and 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.
Location: room A.4.1 C-19
Speaker: Piotr Szewczak (UKSW)
Title: Totally imperfect Menger sets
Abstract: 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 \(d \lt c\) and each totally imperfect Menger set has cardinality less or equal than \(d\).
This is a joint work with Valentin Haberl and 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.