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S. Garc´ıa-Ferreira, R.A. Gonz´ alez-Silva, A.H. Tomita Topological games and product spaces

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S. Garc´ıa-Ferreira, R.A. Gonz´ alez-Silva, A.H. Tomita Topological games and product spaces

Comment.Math.Univ.Carolinae 43,4 (2002) 675-685.

Abstract: In this paper, we deal with the product of spaces which are either G-spaces or Gp-spaces, for some p ∈ ω∗. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are G-spaces, and every Gp-space is a G-space, for every p ∈ ω∗. We prove that if {Xµ : µ < ω1} is a set of spaces whose product X = Q

µ<ω1Xµ is a G-space, then there isA∈[ω1]≤ωsuch that Xµ is countably compact for everyµ∈ω1\A.

As a consequence,Xω1 is aG-space iff Xω1 is countably compact, and if X2c is a G-space, then all powers ofX are countably compact. It is easy to prove that the product of a countable family ofGpspaces is aGp-space, for everyp∈ω∗. For every 1≤n < ω, we construct a spaceX such thatXn is countably compact andXn+1 is not aG-space. Ifp, q∈ω∗ areRK-incomparable, then we construct aGp-space X and aGq-spaceY such thatX×Y is not aG-space. We give an example of two free ultrafilters pand q on ω such that p <RK q, pand q areRF-incomparable, p≈Cq(≤C is theComfortorder onω∗) and there are aGp-spaceX and aGq-space Y whose productX×Y is not aG-space.

Keywords: RF-order,RK-order, Comfort-order,p-limit,p-compact,G-space,Gp- space, countably compact

AMS Subject Classification: Primary 54A35, 03E35; Secondary 54A25

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