Robert Rałowski

# Robert Rałowski: Bernstein set and continuous functions

Tuesday, March 1, 2016 17:15

Room: D1-215

Speaker:
Robert Rałowski

Title: Bernstein set and continuous functions

Abstract. Alexander V. Osipov asked "It is true that for any Bernstein subset $$B\subset \mathbb{R}$$ there are countable many continous functions from $$B$$ to $$\mathbb{R}$$ such that the union of images of $$B$$ is a whole real line $$\mathbb{R}$$". We give the positive answer for this question, but we show that this result is not true for a $$T_2$$ class of functions.

We show some consistency results for completely nonmeasurable sets with respect to $$\sigma$$-ideals of null sets and meager sets on the real line.

These results was obtained commonly with Jacek Cichoń, Michał Morayne and me.

# Robert Rałowski: Two point sets, continuation

Tuesday, March 24, 2015 17:15

Room: D1-215

Speaker:
Robert Rałowski

Title: Two point sets, continuation

Abstract. We will continue discussion started a week ago concerning two point sets. We will give another example of a property of two point set which is consistent with ZFC.

# Robert Rałowski: Cohen indestructible mad families in partial two point sets

Tuesday, March 17, 2015 17:15

Room: D1-215

Speaker:
Robert Rałowski

Title: Cohen indestructible mad families in partial two point sets

Abstract. We discuss on classical construction of Cohen indestructible mad family given by Kenneth Kunen and we apply this method to obtain a partial Cohen indestructible mad family in Baire space as a canonical copy of the real plane.

# Robert Rałowski: On generalized Luzin sets

Tuesday, December 9, 2014 17:15

Room: D1-215

Speaker:
Robert Rałowski

Title: On generalized Luzin sets

Abstract. We will show results obtained together with Sz. Żeberski concerning properties of $$(I,J)$$-Luzin sets (for $$I, J$$ $$\sigma$$-ideals on Polish space). Under some settheoretical assumptions we will construct $$\mathfrak{c}$$ many generalized Luzin sets which are not Borel equivalent. We will also examine some forcing notions which do not kill generalized Luzin sets.

# Robert Rałowski: On m.a.d. $$s_0$$-nonmeasurable sets with a small dominating subfamilies

Tuesday, October 28, 2014 17:15

Room: D1-215

Speaker:
Robert Rałowski

Title: On m.a.d. $$s_0$$-nonmeasurable sets with a small dominating subfamilies

Abstract. We show that $$\mathfrak{d}=\aleph_1$$ implies the existence of maximal familiy of eventually different reals on Baire space which forms a nonmeasurable set with respect to an ideals generated by trees (perfect, Laver or Miller trees for example).

# Robert Rałowski: Nonmeasurability with respect to Marczewski ideal

Tuesday, October 14, 2014 17:15

Room: D1-215

Speaker:
Robert Rałowski

Title: Nonmeasurability with respect to Marczewski ideal

Abstract: Among the others we show relative consistency of ZFC theory with $$\aleph_1< 2^{\aleph_0}$$ and there is a nonmesurable (with respect to ideal generated by complete Laver trees) m.a.d. family $$\mathcal{A}$$ on Baire space $$\omega^\omega$$. In ZFC there is  a subset $$\mathcal{A}’\subseteq \mathcal{A}$$ of size $$\aleph_1$$ unbounded in $$\omega^\omega$$. We show that there is m.a.d. family which is  nonmeasurable with respect to Marczewski ideal.