Arkady Leiderman
Arkady Leiderman: On \(\Delta\)-spaces
04/04/23 11:47
Tuesday, April 4, 2023 17:00
Location: room A.4.1 C-19
Speaker: Arkady Leiderman (Ben-Gurion University of the Negev, Beer Sheva, Israel)
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.
Location: room A.4.1 C-19
Speaker: Arkady Leiderman (Ben-Gurion University of the Negev, Beer Sheva, Israel)
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.