November 2015

Piotr Szewczak: Products of Menger spaces

Tuesday, November 24, 2015 17:15

Room: D1-215

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.

Marcin Michalski: A generalized version of the Rothberger theorem

Tuesday, November 17, 2015 17:15

Room: D1-215

Marcin Michalski

Title: A generalized version of the Rothberger theorem

Abstract. We call a set \(X\) a generalized Luzin set if \(|L\cap M|<|L|\) for every meager set \(M\). Dually, if we replace meager set with a null set, we obtain a definition of a generalized Sierpiński set.

We will show that if \(2^\omega\) is a regular cardinal then for every generalized Luzin set \(L\) and every generalized Sierpiński set \(S\) an algebraic sum \(L+S\) belongs to the Marczewski ideal \(s_0\) (i.e. for every perfect set \(P\) there exists a perfect set \(Q\) such that \(Q\subseteq P\) and \(Q\cap (L+S)=\emptyset\)). To prove the theorem we shall prove and use a generalized version of the Rothberger theorem.

We will also formulate a series of results involving algebraic, topological and measure structure of the real line, that emerged during searching for a proof of the above theorem.