Suppose $X_{\alpha}$, $\alpha\in\Lambda$ is a family of topological spaces and the product set $\Pi_{\alpha\in\Lambda}X_{\alpha}$ is equipped with the box topology. Can we find a generating family of functions for this topology like the product topology generated by the family of projections?
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A similar question (https://math.stackexchange.com/q/3095816/513190) was asked here and attracted some attention, but only a very minor discussion in the comments. It seems like this is a hard question. – Levi Oct 29 '19 at 07:26