April 2023

# Sławomir Solecki: Incomparable Borel linear subspaces of \(\mathbb{R}\) (over \(\mathbb{Q}\))

19/04/23 12:50

Tuesday, April 25, 2023 17:00

*Location:*room**A.4.1 C-19***Sławomir Solecki (Cornell University)*

Speaker:Speaker:

*Title*: Incomparable Borel linear subspaces of \(\mathbb{R}\) (over \(\mathbb{Q}\))*Abstract*: We present a construction of a large family of Borel linear subspaces of \(\mathbb{R}\) (over \(\mathbb{Q}\)), which are incomparable with respect to Borel linear embeddings (over \(\mathbb{Q}\)). A version of this construction answers a question by Frisch and Shinko.# Jonathan Cancino: On nwd-MAD families

18/04/23 14:28

Tuesday, April 18, 2023 17:00

*Location:*room**A.4.1 C-19***Jonathan Cancino (Czech Academy of Sciences)*

Speaker:Speaker:

*Title*: On nwd-MAD families*Abstract*: The cardinal invariant a(nwd) is defined as the minimal cardinality of an uncountable maximal antichain of the power set of the rational modulo the nowhere dense ideal. This cardinal invariant was introduced by J. Steprans, and he proved that in the Laver's model it is \(\omega_1\), and the pseudointersection number p is a lower bound for it. In this talk we will prove some related results, for example, the additivity of the meager ideal is a lower bound for a(nwd), thus improving Steprans theorem, as well as some facts about the structure of nwd-MAD families.# Arkady Leiderman: On \(\Delta\)-spaces

04/04/23 11:47

Tuesday, April 4, 2023 17:00

The talk will be devoted to some results obtained in a recent joint work with Paul Szeptycki (Canada). The aim of this work is to better understand the boundaries of the class $\Delta$, by presenting new examples and counter-examples.

1) We examine when trees considered as topological spaces equipped with the interval topology belong to \(\Delta\).

In particular, we prove that no Souslin tree is a \(\Delta\)-space. Other main results are connected with the study of

2) \(\Psi\)-spaces built on maximal almost disjoint families of countable sets; and

3) Ladder system spaces.

There exists an Isbell-Mrówka \(\Psi\)-space \(X\) (which is in \(\Delta\)) such that one-point extension \(X_p = X \cup \{p\}\) of \(X\) has uncountable tightness at the point \(p\), for some \(p \in \beta(X) \setminus X\).

It is consistent with CH that all ladder system spaces on \(\omega_1\) are \(\Delta\)-spaces.

We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on \(\omega_1\) which is not in \(\Delta\).

[1] Jerzy Kąkol and Arkady Leiderman, A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications, Proc. Amer. Math. Soc., series B, 8 (2021), 86-99.

[2] Jerzy Kąkol and Arkady Leiderman, Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished, Proc. Amer. Math. Soc., series B, (8) (2021), 267-280.

*Location:*room**A.4.1 C-19***Arkady Leiderman (Ben-Gurion University of the Negev, Beer Sheva, Israel)*

Speaker:Speaker:

*Title*: On \(\Delta\)-spaces*Abstract*: \(\Delta\)-spaces have been defined by a natural generalization of a classical notion of \(\Delta\)-sets of reals to Tychonoff topological spaces; moreover, the class \(\Delta\) of all \(\Delta\)-spaces consists precisely of those \(X\) for which the locally convex space \(C_p(X)\) is distinguished. A systematic study of the class \(\Delta\) was originated in my joint papers [1], [2].The talk will be devoted to some results obtained in a recent joint work with Paul Szeptycki (Canada). The aim of this work is to better understand the boundaries of the class $\Delta$, by presenting new examples and counter-examples.

1) We examine when trees considered as topological spaces equipped with the interval topology belong to \(\Delta\).

In particular, we prove that no Souslin tree is a \(\Delta\)-space. Other main results are connected with the study of

2) \(\Psi\)-spaces built on maximal almost disjoint families of countable sets; and

3) Ladder system spaces.

There exists an Isbell-Mrówka \(\Psi\)-space \(X\) (which is in \(\Delta\)) such that one-point extension \(X_p = X \cup \{p\}\) of \(X\) has uncountable tightness at the point \(p\), for some \(p \in \beta(X) \setminus X\).

It is consistent with CH that all ladder system spaces on \(\omega_1\) are \(\Delta\)-spaces.

We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on \(\omega_1\) which is not in \(\Delta\).

[1] Jerzy Kąkol and Arkady Leiderman, A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications, Proc. Amer. Math. Soc., series B, 8 (2021), 86-99.

[2] Jerzy Kąkol and Arkady Leiderman, Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished, Proc. Amer. Math. Soc., series B, (8) (2021), 267-280.