May 2023

# Piotr Szewczak: Totally imperfect Menger sets

31/05/23 10:19

Tuesday, June 6, 2023 17:00

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***Piotr Szewczak (UKSW)*

Speaker:Speaker:

*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.

# Zbigniew Lipecki: How noncompact is the space of Lebesgue measurable sets?

24/05/23 11:41

Tuesday, May 30, 2023 17:00

We show that there exists a sequence of elements of \(\mathfrak M\) such that their mutual distances are > 1/2. It seems to be an open problem whether "1/2" can be replaced here by a bigger constant C. We show that C must be smaller than 9/14. Moreover, we present a version of the problem in terms of binary codes.

*Location:*room**A.4.1 C-19***Zbigniew Lipecki (IM PAN)*

Speaker:Speaker:

*Title*: How noncompact is the space of Lebesgue measurable sets?*Abstract*: The space in question is the space \(\mathfrak M\) of Lebesgue measurable subsets of the unit interval equipped with the usual Fréchet—Nikodym (semi)metric.We show that there exists a sequence of elements of \(\mathfrak M\) such that their mutual distances are > 1/2. It seems to be an open problem whether "1/2" can be replaced here by a bigger constant C. We show that C must be smaller than 9/14. Moreover, we present a version of the problem in terms of binary codes.

# Barnabas Farkas: A tool to avoid some technical forcing arguments when working with the Hechler forcing

17/05/23 14:02

Tuesday, May 23, 2023 17:00

*Location:*room**A.4.1 C-19***Barnabas Farkas (TU Wien)*

Speaker:Speaker:

*Title*: A tool to avoid some technical forcing arguments when working with the Hechler forcing*Abstract*: I'm going to present that virtually every result saying that finite support iterations of the Hechler forcing preserve a cardinal invariant being small and its dual being large can be reduced to a single preservation theorem. In other words, this theorem eliminates the technical forcing arguments from the proofs of these results and reduces them to easy coding exercises.# Damian Sobota: On continuous operators from Banach spaces of Lipschitz functions onto \(c_0\)

16/05/23 14:00

Tuesday, May 16, 2023 17:00

*Location:*room**A.4.1 C-19***Damian Sobota (Kurt Gödel Research Center for Mathematical Logic)*

Speaker:Speaker:

*Title*: On continuous operators from Banach spaces of Lipschitz functions onto \(c_0\)*Abstract*: During my talk I will discuss some of our recent results concerning the existence of continuous operators from the Banach spaces \(\textrm{Lip}_0(M)\) of Lipschitz real-valued functions on metric spaces M onto the Banach space \(c_0\) of sequences converging to \(0\). I will in particular prove that there is always a continuous operator onto \(c_0\) from infinite-dimensional spaces of the form \(\textrm{Lip}_0(C(K))\) or \(\textrm{Lip}_0(\textrm{Lip}_0(M))\). (Based on an ongoing joint work with C. Bargetz and J. Kąkol).