Combinatorics of open covers (V): Pixley–Roy spaces of sets of reals, and ω-covers

Daniels (1988) started an investigation of the duality between selection hypotheses for X⫅R and selection hypotheses for the Pixley–Roy space of X. Daniels, Kunen and Zhou (1994) introduced the “open–open game”. We extend some results of Daniels (1988) by connecting the relevant selection hypotheses with game theory (Theorems 2, 3, 14 and 15) and Ramsey theory (Theorem 10, Corollary 11, Theorem 23 and Corollary 24). Our results give answers to some of the questions asked by Daniels et al. (1994).