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In this paper, we deal with the product of spaces which are either $𝒢$-spaces or ${𝒢}_p$-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 $𝒢$-spaces, and every ${𝒢}_p$-space is a $𝒢$-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 $𝒢$-space, then there is $A\in {\left[{\omega }_1\right]}^{\le \omega }$ such that $X_{\mu }$ is countably compact for every $\mu \in {\omega }_1\setminus A$. As a consequence, $X^{{\omega }_1}$ is a $𝒢$-space iff $X^{{\omega }_1}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$-space, then all...