Olena Karlova: Extension of Borel maps and Borel retracts of topological spaces

Tuesday, December 19, 2017 17:15

Room: D1-215

Speaker:
Olena Karlova

Title: Extension of Borel maps and Borel retracts of topological spaces

Abstract. We will discuss the problem of extension of (dis)continuous maps between topological spaces. Concepts of Baire and Borel retracts of topological spaces will be introduced. Some open problems will be considered.

Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees

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.

Sakae Fuchino: Downward Löwenheim-Skolem Theorems in stationary logic

Tuesday, November 21, 2017 17:15

Room: D1-215

Speaker:
Sakae Fuchino

Title: Downward Löwenheim-Skolem Theorems in stationary logic

Tomasz Natkaniec: Perfectly everywhere surjective but not Jones functions

Tuesday, November 14, 2017 17:15

Room: D1-215

Speaker:
Tomasz Natkaniec

Title: Perfectly everywhere surjective but not Jones functions

Abstract. Given a function \(f:\mathbb{R}\to\mathbb{R}\) we say that

  1. \(f\) is perfectly surjective (\(f\in \mathrm{PES}\)) if \(f[P]=\mathbb{R}\) for every perfect set \(P\);

  2. \(f\) is a Jones function (\(f\in\mathrm{J}\)) if \(C\cap f\neq\emptyset\) for every closed \(C\subset\mathbb{R}^2\) with \(\mathrm{dom}(C)\) of size \(\mathfrak{c}\).


M. Fenoy-Munoz, J.L. Gamez-Merino, G.A. Munoz-Fernandez and E. Saez-Maestro in the paper A hierarchy in the family of real surjective functions [Open Math. 15 (2017), 486--501] asked about the lineability of the set \(\mathrm{PES}\setminus\mathrm{J}\).
Answering this question we show that the class \(\mathrm{PES}\setminus\mathrm{J}\) is \(\mathfrak{c}^+\)-lineable. Moreover, if
\(2^{<\mathfrak{c}}=\mathfrak{c}\) then \(\mathrm{PES}\setminus\mathrm{J}\) is \(2^\mathfrak{c}\)-lineable. We prove also that the additivity number
\(A(\mathrm{PES}\setminus\mathrm{J})\) is between \(\omega_1\) and \(\mathfrak{c}\). Thus \(A(\mathrm{PES}\setminus\mathrm{J})=\mathfrak{c}\) under CH,
however this equality can't be proved in ZFC, because the Covering Property Axiom CPA implies \(A(\mathrm{PES}\setminus\mathrm{J})=\omega_1<\mathfrak{c}\).

The talk is based on the joint paper:
K.C.Ciesielski, J.L. Gamez-Merino, T. Natkaniec, and J.B.Seoane-Sepulveda, On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity, submitted.

Barnabas Farkas: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration

Tuesday, November 7, 2017 17:15

Room: D1-215

Speaker:
Barnabas Farkas (TU Wien)

Title: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration

Abstract. I will present two models concerning interactions between the existence of towers in analytic P-ideals and their cardinal invariants. It is trivial to see that if there is no tower in \(\mathcal{I}\), then \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). I will prove that this implication cannot be reversed no matter the value of \(\mathrm{non}^*(\mathcal{I})\). More precisely, let \(\mathcal{I}\) be an arbitrary tall analytic P-ideal, I will construct the following two models:

Model1 of \(\mathrm{non}^*(\mathcal{I})=\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Small filter iteration.

Model2 of \(\mathrm{non}^*(\mathcal{I})<\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Matrix iteration.

This is a joint work with J. Brendle and J. Verner.