Robert Rałowski

# Robert Rałowski: On \(T_1\)- and \(T_2\)-productable compact spaces

13/03/22 18:05

Tuesday, March 15, 2022 17:00

*Location:*room 605, Mathematical Institute, University of Wroclaw*Robert Rałowski*

Speaker:Speaker:

*Title*: On \(T_1\)- and \(T_2\)-productable compact spaces*Abstract*: We prove that if there exists a continuous surjection from a metric compact space \(X\) onto a product \(X\times T\) where \(T\) is a \(T_1\) second countable topological space which has the cardinality of the continuum then there exists a surjection from \(X\) onto the product \(X\times [0, 1]\) where the interval \([0, 1]\) is equipped with the usual Euclidean topology.# Robert Rałowski: Nonmeasurable unions with respect to analytic families

07/10/20 19:24

Tuesday, October 13, 2020 17:15

1) \((X,\rho)\) is separable metric space,

2) identity \(id:(X,\rho) \to X\) is continuous,

2) every \(\rho\)-Cauchy sequence is converged in \(X.\)

Family \(\mathcal A \subseteq P(X)\) is analytic if

1) \(X\in \mathcal A\)

2) \(\mathcal A\) is closed on intersections

3) each \(A\in{\mathcal A}\) has analytic metric \(\rho\) and for any \(\epsilon>0\) there is a countable cover \(\mathcal U \subseteq \mathcal A\) of \(\mathcal A\) with \(\epsilon\) \(\rho\)-diameter.

We present a theorem that generalizes the well known result obtained by Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski about nonmeasurable unions.

Theorem. Let \({\mathcal A}\) be an analytic family of Hausdorff space \(X\), any \(I\) \(\sigma\)-ideal of \(X.\) If \( J\subseteq I\) is point-finite family such that \(\bigcup J \notin I\) then there is a subfamily \(J' \subseteq J\) and \(A\in {\mathcal A}\) such that

1) \(A\cap \bigcap J' \notin I\)

2) for every \(A' \in {\mathcal A}\) if \(A' \subseteq A\cap \bigcup J'\) then \(A' \in I.\)

We show that the above Theorem implies the Theorem on nomeasurabie unions with respect to tree ideals like Marczeski ideal \(s_0\) for example.

Moreover, the above Theorem implies theorem on nonmeasurable unions with respect to \(\sigma\)-ideals which has Marczewski-Burstin representation.

The last mentioned result gives a theorem about nonmeasurable unions with respect to the ideal of Ramsey-null set in Ramsey space with Ellentuck topology.

The talk is based on a joint work with Taras Banakh and Szymon Żeberski.

*Location:***Zoom.us**: if you want to participate please contact organizers*Robert Rałowski*

Speaker:Speaker:

*Title*: Nonmeasurable unions with respect to analytic families*Abstract*: We say that metric \(\rho\) is analytic on Hausdorff topological space if1) \((X,\rho)\) is separable metric space,

2) identity \(id:(X,\rho) \to X\) is continuous,

2) every \(\rho\)-Cauchy sequence is converged in \(X.\)

Family \(\mathcal A \subseteq P(X)\) is analytic if

1) \(X\in \mathcal A\)

2) \(\mathcal A\) is closed on intersections

3) each \(A\in{\mathcal A}\) has analytic metric \(\rho\) and for any \(\epsilon>0\) there is a countable cover \(\mathcal U \subseteq \mathcal A\) of \(\mathcal A\) with \(\epsilon\) \(\rho\)-diameter.

We present a theorem that generalizes the well known result obtained by Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski about nonmeasurable unions.

Theorem. Let \({\mathcal A}\) be an analytic family of Hausdorff space \(X\), any \(I\) \(\sigma\)-ideal of \(X.\) If \( J\subseteq I\) is point-finite family such that \(\bigcup J \notin I\) then there is a subfamily \(J' \subseteq J\) and \(A\in {\mathcal A}\) such that

1) \(A\cap \bigcap J' \notin I\)

2) for every \(A' \in {\mathcal A}\) if \(A' \subseteq A\cap \bigcup J'\) then \(A' \in I.\)

We show that the above Theorem implies the Theorem on nomeasurabie unions with respect to tree ideals like Marczeski ideal \(s_0\) for example.

Moreover, the above Theorem implies theorem on nonmeasurable unions with respect to \(\sigma\)-ideals which has Marczewski-Burstin representation.

The last mentioned result gives a theorem about nonmeasurable unions with respect to the ideal of Ramsey-null set in Ramsey space with Ellentuck topology.

The talk is based on a joint work with Taras Banakh and Szymon Żeberski.

# Robert Rałowski: Mycielski among trees - nonstandard proofs, part 2

04/11/19 17:11

Tuesday, November 5, 2019 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: Mycielski among trees - nonstandard proofs, part 2*Abstract*. We present proofs of Mycielski like Theorem for sigma ideal of meager subsets of Baire space and Egglestone like Theorem. In both proofs we use Schoenfield Absolutness Theorem. In Mycielski Theorem we replace the term perfect set by slalom perfect set what is some strengthen of the classical version. Results are from common paper with Marcin Michalski and Szymon Żeberski.# Robert Rałowski: Mycielski among trees - nonstandard proofs

23/10/19 22:48

Tuesday, October 29, 2019 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: Mycielski among trees - nonstandard proofs*Abstract*. We present proofs of Mycielski like Theorem for sigma ideal of meager subsets of Baire space and Egglestone like Theorem. In both proofs we use Schoenfield Absolutness Theorem. In Mycielski Theorem we replace the term perfect set by slalom perfect set what is some strengthen of the classical version. Results are from common paper with Marcin Michalski and Szymon Żeberski.# Robert Rałowski: Images of Bernstein sets via continuous functions

07/11/18 06:43

Tuesday, November 13, 2018 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: Images of Bernstein sets via continuous functions*Abstract*. We examine images of Bernstein sets via continuous mappings. Among other results we prove that there exists a continuous function \(f:\mathbb{R}\to\mathbb{R}\) that maps every Bernstein subset of \(\mathbb{R}\) onto the whole real line. This gives the positive answer to a question of Osipov. This talk is based upon joint paper with Jacek Cichoń and Michał Morayne.# Robert Rałowski: Bernstein set and continuous functions

25/02/16 15:41

Tuesday, March 1, 2016 17:15

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.

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*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

23/03/15 20:46

Tuesday, March 24, 2015 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*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

11/03/15 17:35

Tuesday, March 17, 2015 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*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

05/12/14 18:34

Tuesday, December 9, 2014 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*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

27/10/14 16:02

Tuesday, October 28, 2014 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*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

09/10/14 17:51

Tuesday, October 14, 2014 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*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.