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Topological games and product spaces. (English) Zbl 1090.54005
Products of \({\mathcal G}\)-spaces and of \({\mathcal G}_p\)-spaces are investigated. \({\mathcal G}\)-spaces (containing countably compact spaces) and \({\mathcal G}_p\)-spaces (containing \(p\)-compact spaces), for \(p\in \omega ^*\), are defined by special games. From results on infinite products: (1) If \(\Pi \{X_\alpha :\alpha \in \omega _1\}\) is a \({\mathcal G}\)-space (or a \({\mathcal G}_p\)-space) then all \(X_\alpha \), except countably many, are countably compact (or \(p\)-compact, resp.). (2) Every power of \(X\) is a \({\mathcal G}_p\)-space iff \(X\) is \(p\)-compact, every power of \(X\) is a \({\mathcal G}\)-space iff \(X\) is \(p\)-compact for some \(p\). Various results on Rudin-Frolík and Rudin-Keisler orders on \(\omega ^*\) are used to construct various situations for finite products of \({\mathcal G}\)-spaces or \({\mathcal G}_p\)-spaces (e.g., using a Frolík’s approach, for every \(1\leq n<\omega \) there is \(X\) with \(X^n\) countably compact and \(X^{n+1}\) not a \({\mathcal G}\)-space, or if \(p,q\) are not RK-comparable, there are a \({\mathcal G}_p\)-space \(X\) and a \({\mathcal G}_q\)-space \(Y\) with \(X\times Y\) not a \({\mathcal G}\)-space).

MSC:
54B10 Product spaces in general topology
54D99 Fairly general properties of topological spaces
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