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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
##### Keywords:
product of spaces; $$\mathcal G$$-spaces; ultrafilters
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