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Keywords:
$RF$-order; $RK$-order; {\it Comfort}-order; $p$-limit; $p$-compact; $\Cal G$-space; $\Cal G_p$-space; countably compact
$RF$-order; $RK$-order; {\it Comfort}-order; $p$-limit; $p$-compact; $\Cal G$-space; $\Cal G_p$-space; countably compact
Summary:
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.
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