S. Garc\'{\i }a-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 $\Cal G$-spaces or $\Cal G_p$-spaces, for some $p \in \omega ^*$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are ${\cal G}$-spaces, and every $\Cal G_p$-space is a $\Cal G$-space, for every $p \in \omega ^*$. We prove that if $\{ X_\mu : \mu < \omega _1 \}$ is a set of spaces whose product $X= \prod _{\mu < \omega _1}X_ \mu $ is a $\Cal G$-space, then there is $A \in [\omega _1]^{\leq \omega }$ such that $X_\mu $ is countably compact for every $\mu \in \omega _1 \setminus A$. As a consequence, $X^{\omega _1}$ is a $\Cal G$-space iff $X^{\omega _1}$ is countably compact, and if $X^{2^{\frak c}}$ is a $\Cal G$-space, then all powers of $X$ are countably compact. It is easy to prove that the product of a countable family of $\Cal G_p$ spaces is a $\Cal G_p$-space, for every $p \in \omega ^*$. For every $1 \leq n < \omega $, we construct a space $X$ such that $X^n$ is countably compact and $X^{n+1}$ is not a $\Cal G$-space. If $p, q \in \omega ^*$ are $RK$-incomparable, then we construct a $\Cal G_p$-space $X$ and a $\Cal G_q$-space $Y$ such that $X \times Y$ is not a $\Cal G$-space. We give an example of two free ultrafilters $p$ and $q$ on $\omega $ such that $p <_{RK} q$, $p$ and $q$ are $RF$-incomparable, $p \approx _C q$ ($\leq _C$ is the {Comfort} order on $\omega ^*$) and there are a $\Cal G_p$-space $X$ and a $\Cal G_q$-space $Y$ whose product $X \times Y$ is not a $\Cal G$-space.

Keywords: $RF$-order, $RK$-order, {Comfort}-order, $p$-limit, $p$-compact, ${\cal G}$-space, ${\cal G}_p$-space, countably compact
AMS Subject Classification: Primary 54A35, 03E35; Secondary 54A25