About the Box and Product topologies

Solution 1:

I'm not sure if I understand your question but what you're saying is correct. When considering $\prod_i X_i$, we can endow it either with the box topology or the product topology. Unless the indexing set is finite, these will give you different topologies.

The box topology is indeed characterised by the fact that the interior of a product $\prod_i A_i \subseteq \prod_i X_i$ will be $\prod_i \operatorname{Int}(A_i)$. This is not the case in the product topology. Indeed, if infinitely many of the $A_i$ are proper subspaces of $X_i$, then $\prod_i \operatorname{Int}(A_i)$ will not even be an open subset with respect to the product topology.

In the case that infinitely many of the $A_i$ are proper subspaces of $X_i$, do you have an idea what the interior of $\prod_i A_i$ will be with respect to the product topology? The answer may surprise you!