Archives of: Wrocław University of Technology

Ziemowit Kostana: Non-measurability of algebraic sum

Sunday, September 17, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Ziemowit Kostana (University of Warsaw)

Title: Non-measurability of algebraic sum

Abstract:

Consider following problems:
1. If A is meagre (null) subset of real line, does there necessarily exist set B such that algebraic sum A+B doesn’t have Baire property (is non-measurable)?
2. If A is meagre (null) subset of real line, does there necessarily exist non-meagre (non-null) additive subgroup, disjoint with some translation of A?

It is not hard to prove that positive answer to 2. implies positive answer to 1, both for measure and category.
We answer 2. affirmatively for category, while version for measure turns out to be independent of ZFC. The latter was essentially proved last year
by A. Rosłanowski and S. Shelah. Both results holds for Cantor space with coordinatewise addition mod. 2 as well.

Aleksander Cieślak: Ideals of subsets of plane

Sunday, September 10, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wrocław University of Technology)

Title: Ideals of subsets of plane

Abstract:

For given two ideals $I$ and $J$ of subsets of Polish space $X$ we define a Fubini product $I\times J$ as all these subsets of plane $X^2$ which can be covered by a Borel set $B$ such that $I$-almost all its vertical sections are $J$-small. We will investigate how properties of factors influence properties of product.

Aleksander Cieślak: Cohen-stable families of subsets of integers

Tuesday, June 13, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wroclaw University of Science and Technology)

Title: Cohen-stable families of subsets of integers

Abstract:

A mad family is Cohen-stable if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-stable. We will find condition necessary and sufficient for mad family to be Cohen-unstabe and investigate when such family exist.

Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jarosław Swaczyna (Lodz University of Technology)

Title: Haar-small sets

Abstract:

In locally compact Polish groups there is a very natural $\sigma$-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.

Joanna Jureczko: Some remarks on Kuratowski partitions, new results

Tuesday, May 9, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Joanna Jureczko (Cardinal Stefan Wyszynski University in Warsaw)

Title: Some remarks on Kuratowski partitions, new results

Abstract:

K. Kuratowski in 1935 posed the problem whether a function $f \colon X \to Y$ from a completely metrizable space $X$ to a metrizable space $Y$ is continuous apart from a meager set.
This question is equivalent to the question about the existence of so called a Kuratowski partition, i. e. a partition $\mathcal{F}$ of a space $X$ into meager sets such that $\bigcup \mathcal{F}’$ for any $\mathcal{F}’ \subset \mathcal{F}$.
With any Kuratowski partition we may associate a $K$-ideal, i.e. an ideal of the form
$$I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A}F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.$$
It would seem that the information about $I_{\mathcal{F}}$ would give us full information about the ideal and the world in which it lives.
My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier.
During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz – Kunen Theorem (1987) on the existence of measurable cardinals.

Marcin Michalski: Luzin’s theorem

Tuesday, April 25, 2017, 17:15
Wrocław University of Technology, 215 D-1

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

Title: Luzin’s theorem

Abstract:

In 1934 Nicolai Luzin proved that each subset of the real line can be decomposed into two full subsets with respect to ideal of measure or category. We shall present the proof of this result partially decoding his work and we will also briefly discuss possible generalizations.

Aleksander Cieślak: Indestructible tower

Tuesday, April 11, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wroclaw University of Science and Technology)

Title: Indestructible tower

Abstract:

Following the Kunen’s construction of m.a.d. family which is indestructible over adding $\omega_2$ Cohen reals we provide analogous construction for indestructibe tower.

Judyta Bąk: Domain theory and topological games

Tuesday, March 28, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Judyta Bąk (University of Silesia)

Title: Domain theory and topological games

Abstract:

Domain is a partially ordered set, in which there was introduced some specific relation. We say that a space is domain representable if it is homeomorphic to a space of maximal elements of some domain. In 2015 W. Fleissner and L. Yengulalp introduced a notion of $\pi$–domain representable space, which is analogous of domain representable. We prove that a player $\alpha$ has a winning strategy in the Banach–Mazur game on a space $X$ if and only if $X$ is countably $\pi$–domain representable. We give an example of countably $\pi$–domain representable space, which is not $\pi$–domain representable.

Piotr Szewczak: The Scheepers property and products of Menger spaces

Tuesday, March 14, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Szewczak (Cardinal Stefan Wyszynski University in Warsaw)

Title: The Scheepers property and products of Menger spaces

Abstract:

A topological space $X$ is Menger if for every sequence of open covers $\mathcal{O}_1, \mathcal{O}_2,\ldots $ of the space $X$, there are finite subfamilies $\mathcal{F}_1\subseteq \mathcal{O}_1, \mathcal{F}_2\subseteq\mathcal{O}_2,\ldots $ such that their union is a cover of $X$. If, in addition, for every finite subset $F$ of $X$ there is a natural number $n$ with $F\subseteq\bigcup\mathcal{F}_n$, then the space $X$ is Scheepers. The above properties generalize $\sigma$-compactness, and Scheepers’ property is formally stronger than Menger’s property. It is consistent with ZFC that these properties are equal.

One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely
combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces

This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).

Aleksander Cieślak: Strongly meager sets and subsets of the plane

Tuesday, December 20, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wrocław University of Science and Technology)

Title: Strongly meager sets and subsets of the plane

Abstract:

We will show some results proved by J. Pawlikowski in “Strongly meager sets and subsets of the plane”.