October 2020

# Paweł Krupski: The complexity of homogeneous continua

27/10/20 16:02

Tuesday, November 3, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Paweł Krupski*

Speaker:Speaker:

*Title*: The complexity of homogeneous continua*Abstract*: I will show that the family of all homogeneous continua in the hyperspace of all subcontinua of the cube \(I^n, n=2,3,\ldots ,\omega\), is analytic and contains a topological copy of the linear space \(c_0=\{(x_k)\in {\mathbb{R}}^\omega: \lim x_k=0\}\) as a closed subset. A historical background will also be sketched.# Antonio Aviles: Amalgamation of measures and Banach lattices

21/10/20 15:10

Tuesday, October 27, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Antonio Aviles (University of Murcia)*

Speaker:Speaker:

*Title*: Amalgamation of measures and Banach lattices*Abstract*: Given two measures that coincide on the intersection of their domains, can we find a measure that is a common extension of those two? Kellerer's results on marginal measures constitute an important partial positive answer. We will see how this is connected to some basic properties of the category of Banach lattices, like amalgamation and existence of injective objects. Joint work with Pedro Tradacete.# Jan van Mill: Splitting Tychonoff cubes into homeomorphic and homogeneous parts (and more)

14/10/20 07:59

Tuesday, October 20, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Jan van Mill (University of Amsterdam)*

Speaker:Speaker:

*Title*: Splitting Tychonoff cubes into homeomorphic and homogeneous parts (and more)*Abstract*: We prove (among other things) that if \(X\) is the Tychonoff cube of weight \(\tau\), where \(\tau\) is uncountable, and \(\mathcal{E}\) is a cover of \(X\) by subspaces each homeomorphic to a topological group, then \(|\mathcal{E}|\ge \tau^+\).# 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.