Tuesday, November 17, 2015, 17:15
Wrocław University of Technology, 215 D-1
Speaker: Marcin Michalski (Wrocław University of Technology)
Title: A generalized version of the Rothberger theorem
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.