March 2017

Judyta Bąk: Domain theory and topological games

Tuesday, March 28, 2017 17:15

Room: D1-215

Judyta Bąk

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

Tuesday, March 14, 2017 17:15

Room: D1-215

Piotr Szewczak

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).