Combinatorics of open covers. V: Pixley-Roy spaces of sets of reals, and \(\omega\)-covers
In earlier work [``Combinatorics of open covers'', Parts I--IV, see ibid. 83, No. 1, 63-75 (1998; Zbl 0947.91016)] of the author were investigated some of the combinatorial consequences associated with the fact that certain topologically significant families of sets satisfy certain selection hypotheses. Here these are developed further. For collections of subsets \({\mathcal A}\) and \({\mathcal B}\) of an infinite set, the two selection hypotheses \(S_1({\mathcal A},{\mathcal B})\) are defined as follows: for every sequence \((O_n: n\in \mathbb{N})\) of elements of \({\mathcal A}\) there is a sequence \((T_n: n\in \mathbb{N})\) such that for each \(n\), \(T_n\in O_n\) and \(\{T_n\}\) is an element of \({\mathcal B}\). In this work the author extends by connecting the relevant selection hypotheses with game theory and Ramsey theory, the investigations of Daniels, Kunen and Zhou about duality between selection hypotheses for \(X\subset R\) and selection hypotheses for the Pixley-Roy space of \(X\).
scientific article; zbMATH DE number 1422594