Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees
22/11/17 10:49
Tuesday, November 28, 2017 17:15
Room: D1-215
Speaker: Marcin Michalski
Title: Bernstein, Luzin and Sierpiński meet trees
Abstract. In [2] we have proven that if \(\mathfrak{c}\) is a regular cardinal number, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to Marczewski ideal \(s_0\). We will generalize this result for other tree ideals - \(m_0\) and \(l_0\) - using some lemmas on special kind of fusion sequences for trees of respective type.
Let us introduce a following notion. Let \(\mathbb{X}\) be a set of trees.
Definition. We call a set \(B\) a \(\mathbb{X}\)-Bernstein set, if for each \(X\in\mathbb{X}\) we have \([X]\cap B\neq\emptyset\).
We shall explore this notion for various set of trees, including Sacks, Miller and Laver trees, with the support of technics developed in [1].
[1] Brendle J., Strolling through paradise, Fundamenta Mathematicae, 148 (1995), pp. 1-25.
[2] Michalski M., Żeberski Sz., Some properties of I-Luzin, Topology and its Applications, 189 (2015), pp. 122-135.
Room: D1-215
Speaker: Marcin Michalski
Title: Bernstein, Luzin and Sierpiński meet trees
Abstract. In [2] we have proven that if \(\mathfrak{c}\) is a regular cardinal number, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to Marczewski ideal \(s_0\). We will generalize this result for other tree ideals - \(m_0\) and \(l_0\) - using some lemmas on special kind of fusion sequences for trees of respective type.
Let us introduce a following notion. Let \(\mathbb{X}\) be a set of trees.
Definition. We call a set \(B\) a \(\mathbb{X}\)-Bernstein set, if for each \(X\in\mathbb{X}\) we have \([X]\cap B\neq\emptyset\).
We shall explore this notion for various set of trees, including Sacks, Miller and Laver trees, with the support of technics developed in [1].
[1] Brendle J., Strolling through paradise, Fundamenta Mathematicae, 148 (1995), pp. 1-25.
[2] Michalski M., Żeberski Sz., Some properties of I-Luzin, Topology and its Applications, 189 (2015), pp. 122-135.