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 {\it 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.
@article{119356,
author = {Salvador Garc\'\i a-Ferreira and R. A. Gonz\'alez-Silva and Artur Hideyuki Tomita},
title = {Topological games and product spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {675-685},
zbl = {1090.54005},
mrnumber = {2045789},
language = {en},
url = {http://dml.mathdoc.fr/item/119356}
}
García-Ferreira, Salvador; González-Silva, R. A.; Tomita, Artur Hideyuki. Topological games and product spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 675-685. http://gdmltest.u-ga.fr/item/119356/
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