Marcin Michalski: A generalized version of the Rothberger theorem
16/11/15 16:59
Tuesday, November 17, 2015 17:15
Room: D1-215
Speaker: 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.
Room: D1-215
Speaker: 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.