Reznichenko families of trees and their applications (Q2518769)

A hereditary family \(\mathcal{F}\) of subsets of a set \(A\) is one satisfying the condition: if \(F\in\mathcal{F}\) and \(G\subset F\) then \(G\in\mathcal{F}\) as well. Let \(\mathcal{F}\) be a hereditary family and \(A=\cup \mathcal{F}\). A Reznichenko family determined by \((A,\mathcal{F})\) is a family of trees \(\{T_a:a\in A\}\) with the following properties: a) Setting \(\mathbf{m}=|\bigcup_{a\in A}T_a|\), if there exists an infinite \(F\in\mathcal{F}\) then \(\mathrm{cf}(\mathbf{m})\geq\omega_1\). b) For every \(a\in A,\;T_a\) is a tree of height \(\omega\) and root \(a\) and moreover \(T_a\cap A=\{a\}\). c) If \(a_1,a_2\in A\), \(a_1\neq a_2\) and \(I_1,I_2\) are segments of \(T_{a_1},T_{a_2}\) respectively, then \(|I_1\cap I_2|\leq 1\). d) If \(F\in\mathcal{F}\) and \(I_a\) are pairwise disjoint finite initial segments of \(T_a, a\in F\), then \(|\cap_{a\in F}\mathrm{imsuc}_a(\max I_a)|=\mathbf{m}\). (Here \(\mathrm{imsuc}_a(t)\) is the set of immediate successors of \(t\) in \(T_a\). e) For every \(t\in\cup_{a\in A}T_a\), \(\{a\in A:t\in T_a\}\in \mathcal{F}\). In the above definition, a segment \(I\) in a tree \(T\) is a totally ordered subset of \(T\) such that if \(t_1,t_2\in I\) and \(t_1\leq t\leq t_2\), then \(t\in I\). A segment \(I\) is initial if for every \(t_1\in I\) and \(t\leq t_1\), we have \(t\in I\) as well. This concept was introduced in [\textit{S. A.~Argyros, A. D.~Arvanitakis} and \textit{S. K.~Mercourakis}, Topology Appl. 155, 1737--1755 (2008; Zbl 1158.46011)]. The present paper is devoted to a further and more systematic study of Reznichenko families of trees. Using these families, the authors present new nontrivial examples of Talagrand, Gul'ko and Corson compacta. A new proof is given of the \(K_{\sigma\delta}\) property for \(\mathcal{K}\)-analytic Banach spaces with an unconditional basis. The proof is considerably simpler than the one presented in the above mentioned paper.