Marcin Michalski

# Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees

22/11/17 10:49

Tuesday, November 28, 2017 17:15

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*Marcin Michalski*

Speaker:Speaker:

*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.

# Marcin Michalski: Luzin's theorem

24/04/17 08:28

Tuesday, April 25, 2017 17:15

*Room:*D1-215*Marcin Michalski*

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*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.# Marcin Michalski: Decomposing the real line into Borel sets closed under addition

13/12/16 00:07

Tuesday, December 13, 2016 17:15

*Room:*D1-215*Marcin Michalski*

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*Title*: Decomposing the real line into Borel sets closed under addition*Abstract*. We will show some results proved by M. Elekes and T.Keleti in "Decomposing the real line into Borel sets closed under addition".# Marcin Michalski: Universal sets for bases of \(\sigma\)-ideals

04/11/16 16:15

Tuesday, November 8, 2016 17:15

*Room:*D1-215*Marcin Michalski*

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*Title*: Universal sets for bases of \(\sigma\)-ideals*Abstract*. We shall construct universal sets of possibly low Borel rank for classic \(\sigma\)-ideals of sets: \(\mathcal{N}\)-family of measure zero sets, \(\mathcal{M}\)-family of meager sets, \(\mathcal{M}\cap\mathcal{N}\) and \(\mathcal{E}\). We will also discuss briefly cases of other ideals.# Marcin Michalski: On some properties of sigma-ideals

17/10/16 22:50

Tuesday, October 18, 2016 17:15

*Room:*D1-215*Marcin Michalski*

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*Title*: On some properties of sigma-ideals*Abstract*. We shall consider a couple of properties of \(\sigma\)-ideals and relations between them. Namely we will prove that \(\mathfrak c\)-cc \(\sigma\)-ideals are tall, Weaker Smital Property implies that every Borel \(\mathcal{I}\)-positive set contains a witness for non(\(\mathcal{I}\)) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of \(\mathcal{I}\)-Luzin sets and prove that the ideal \([\mathbb R]^{\leq\omega}\) does not posses the Fubini Property using some interesting lemma about perfect sets.# Marcin Michalski: A generalized version of the Rothberger theorem

16/11/15 16:59

Tuesday, November 17, 2015 17:15

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*Marcin Michalski*

Speaker:Speaker:

*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.

# Marcin Michalski: Avoiding rational distances

09/03/15 10:48

Tuesday, March 10, 2015 17:15

*Room:*D1-215*Marcin Michalski*

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*Title*: Avoiding rational distances*Abstract*. We shall present results obtained by Ashutosh Kumar in paper "Avoiding rational distances". The author showed that for any set of reals X of positive outer measure there exists a subset Y of X such that Y has the same outer measure and the distance between any distinct points of Y is irrational. We will also discuss briefly the case of higher dimensions.# Marcin Michalski: I-Luzin sets

23/01/15 14:44

Tuesday, January 27, 2015 17:15

*Room:*D1-215*Marcin Michalski*

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*Title*: I-Luzin sets*Abstract*. We will present some results obtained with Szymon Żeberski involving I-Luzin sets in Euclidean spaces. We shall construct a quite decent (non-trivial, possesing Borel base and translation invariant) sigma-ideal I of sets such that there exists an I-measurable I-Luzin set. We give also sufficient condition of I-nonmeasurability of I-Luzin sets involving Smital Property (precisely- it's weaker version). We also discuss briefly Stenihaus and Smital Properties of Fubini product of sigma-ideals.# Marcin Michalski: Luzin and Sierpiński sets, some nonmeasurable subsets of the plane

16/10/14 19:06

Tuesday, October 21, 2014 17:15

*Room:*D1-215*Marcin Michalski*

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*Title:*Luzin and Sierpiński sets, some nonmeasurable subsets of the plane*Abstract:*We shall introduce some nonmeasurable and completely nonmeasurable subsets of the plane with various additional properties, e.g. being Hamel basis, intersecting each line in a strong Luzin/Sierpiński set. Also some additive properties of Luzin and Sierpiński sets and their generalization, \(\mathcal{I}\)-Luzin sets, on the line are investigated.