March 2024

# Tomasz Żuchowski: The Nikodym property and filters on \(\omega\). Part I

25/03/24 11:14

Tuesday, March 26, 2024 17:15

In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.

*Location:*A.4.1 C-19*Tomasz Żuchowski*

Speaker:Speaker:

*Title*: The Nikodym property and filters on \(\omega\). Part I*Abstract*: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.

# Piotr Szewczak: Perfectly meager sets in the transitive sense and the Hurewicz property

18/03/24 11:11

Tuesday, March 19, 2024 17:15

This is a joint work with Tomasz Weiss 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:*A.4.1 C-19*Piotr Szewczak (UKSW)*

Speaker:Speaker:

*Title*: Perfectly meager sets in the transitive sense and the Hurewicz property*Abstract*: We work in the Cantor space with the usual group operation +. A set X is perfectly meager in the transitive sense if for any perfect set P there is an F-sigma set F containing X such that for every point t the intersection of t+F and P is meager in the relative topology of P. A set X is Hurewicz if for any sequence of increasing open covers of X one can select one set from each cover such that the chosen sets formulate a gamma-cover of X, i.e., an infinite cover such that each point from X belongs to all but finitely many sets from the cover. Nowik proved that each Hurewicz set which cannot be mapped continuously onto the Cantor set is perfectly meager in the transitive sense. We answer a question of Nowik and Tsaban, whether of the same assertion holds for each Hurewicz set with no copy of the Cantor set inside. We solve this problem, under CH, in the negative.This is a joint work with Tomasz Weiss 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

# Agnieszka Widz: Random graph

05/03/24 07:47

Tuesday, March 5, 2024 17:15

*Location:*A.4.1 C-19*Agnieszka Widz*

Speaker:Speaker:

*Title*: Random graph*Abstract*: The Random Graph can be generated almost surely by connecting vertices with a fixed probability \(p\in(0,1)\), independently of other pairs. In my talk, I will recall the construction and explore interesting properties of the Random Graph, investigating the impact of varying probabilities for each edge. Specifically, I will characterize sequences \((p_n)_{n\in\mathbb{N}}\) for which there exists a bijection \(f\) between pairs of vertices in \(\mathbb{N}\), such that if we connect vertices \(v\) and \(w\) with probability \(p_{f(\{v,w\})}\), the Random Graph emerges almost surely.