On the product of almost discrete Grothendieck spaces

Abstract: A topological space $X$ is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X=\prod X_i$ of almost discrete spaces $X_i$ the space $C_p(X)$ of continuous real-valued functions with the topology of pointwise convergence is a $\mu$-space if, and only if, $X$ is a weak $q$-space if, and only if, $t(X)=\omega$ if, and only if, $X$ is functionally generated by the family of all its countable subspaces. This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of $ZFC$, obtained by adding one Cohen real, there are Grothendieck spaces $X$ and $Y$ such that $X\times Y$ is not weakly Grothendieck space. In $(PFA)$: the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.