May 2017

# Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017 17:15

Room: D1-215

Speaker:
Jarosław Swaczyna

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

Room: D1-215

Speaker:
Joanna Jureczko

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.