Combinatorics of open covers. V: Pixley-Roy spaces of sets of reals, and \(\omega\)-covers
type: Item
instance of: scholarly article
title: Combinatorics of open covers. V: Pixley-Roy spaces of sets of reals, and \(\omega\)-covers
zbMATH Open document ID: 0960.91018
DOI: 10.1016/S0166-8641(98)00141-2
author: Marion Scheepers
published in: Topology and its Applications
publication date: 2000-03-23T00:00:00Z
review text: 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 \(X\).
Mathematics Subject Classification ID: 91A44
Mathematics Subject Classification ID: 91A10
Mathematics Subject Classification ID: 91A05
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 1422594
zbMATH Keywords: infinite games
zbMATH Keywords: Ramsey's theorem
zbMATH Keywords: selection hypotheses
zbMATH Keywords: Rothberger property
reviewed by: A. A. Rakhimov
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00141-2
OpenAlex ID: W1983220719
description: scientific article; zbMATH DE number 1422594