Piotr Szewczak: The Scheepers property and products of Menger spaces

Tuesday, March 14, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Szewczak (Cardinal Stefan Wyszynski University in Warsaw)

Title: The Scheepers property and products of Menger spaces


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).

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