March 2017

# Judyta Bąk: Domain theory and topological games

22/03/17 09:21

Tuesday, March 28, 2017 17:15

*Room:*D1-215*Judyta Bąk*

Speaker:Speaker:

*Title*: Domain theory and topological games*Abstract*. Domain is a partially ordered set, in which there was introduced some specific relation. We say that a space is domain representable if it is homeomorphic to a space of maximal elements of some domain. In 2015 W. Fleissner and L. Yengulalp introduced a notion of \(\pi\)-domain representable space, which is analogous of domain representable. We prove that a player \(\alpha\) has a winning strategy in the Banach--Mazur game on a space \(X\) if and only if \(X\) is countably \(\pi\)-domain representable. We give an example of countably \(\pi\)-domain representable space, which is not \(\pi\)-domain representable.# Piotr Szewczak: The Scheepers property and products of Menger spaces

10/03/17 08:01

Tuesday, March 14, 2017 17:15

One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely

combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces

This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).

*Room:*D1-215*Piotr Szewczak*

Speaker:Speaker:

*Title*: The Scheepers property and products of Menger spaces*Abstract*. A topological space \(X\) is Menger if for every sequence of open covers \(\mathcal{O}_1, \mathcal{O}_2,\ldots \) of the space \(X\), there are finite subfamilies \(\mathcal{F}_1\subseteq \mathcal{O}_1,\ \mathcal{F}_2\subseteq\mathcal{O}_2,\ldots \) such that their union is a cover of \(X\). If, in addition, for every finite subset \(F\) of \(X\) there is a natural number \(n\) with \(F\subseteq\bigcup\mathcal{F}_n\), then the space \(X\) is Scheepers. The above properties generalize \(\sigma\)-compactness, and Scheepers’ property is formally stronger than Menger’s property. It is consistent with ZFC that these properties are equal.One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely

combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces

This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).