# Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees

Tuesday, November 28, 2017, 17:15
Wrocław University of Science and Technology, 215 D-1

Speaker: Marcin Michalski (Wrocław University of Science and Technology)

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.