Piotr Szewczak

# Piotr Szewczak: The Scheepers property and products of Menger spaces

Tuesday, March 14, 2017 17:15

Room: D1-215

Speaker:
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).

# Piotr Szewczak: Products of Menger spaces

Tuesday, November 24, 2015 17:15

Room: D1-215

Speaker:
Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw); Coauthor: Boaz Tsaban (Bar-Ilan University, Israel)

Title: Products of Menger spaces

Abstract. A topological space $$X$$ is Menger if for every sequence of open covers $$O_1, O_2, \ldots$$ there are finite subfamilies $$F_1$$ of $$O_1$$, $$F_2$$ of $$O_2$$, . . . such that their union is a cover of $$X$$. The above property generalizes sigma-compactness.

One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.