Let $\{X_{\alpha}\}$ be an uncountable collection of topological spaces indexed by the set $J$. For the space $\prod_{\alpha}X_{\alpha}$, consider the topology $\tau$ generated by the basis
$$\mathcal{B}=\left\{\left(\prod_{\alpha\in S}U_{\alpha}\right)\times\left(\prod_{\beta\notin S}X_{\beta}\right): U_{\alpha}\text{ open in }X_{\alpha},S\subseteq J\right\}.$$
If $S$ is arbitrary, then $\tau$ is the box topology. If $S$ is finite, then $\tau$ is the product topology. What if we allow $S$ to be countable? Has this topology been studied at all? Does it have a use somewhere?