May 2017

# Jarosław Swaczyna: Haar-small sets

18/05/17 10:55

Tuesday, May 23, 2017 17:15

*Room:*D1-215*Jarosław Swaczyna*

Speaker:Speaker:

*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

05/05/17 22:15

Tuesday, May 9, 2017 17:15

\(I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A} F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.\)

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.

*Room:*D1-215*Joanna Jureczko*

Speaker:Speaker:

*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.