Product and Box Topologies

Fix a set $X=\prod_{\alpha\in I} X_\alpha$ where $I$ is infinite, and consider the two topologies on it.

The box topology is finer than the product topology because if $U_\alpha\subset X_\alpha$ is a proper subset, then $U=\prod_\alpha U_\alpha$ will be open in the box topology (and is basic), but it is not open in the product topology. Such a set cannot be open in the product topology, because every open set in the product topology is the union basic open sets, each of which has $U_\alpha=X_\alpha$ for all but finitely many terms. In particular, this means any open set of $X$ in the product topology must contain a subset of the form $\prod_\alpha U_\alpha$ where $U_\alpha=X_\alpha$ for almost all $\alpha$.