# Are there useful topologies on Cartesian products “in between” the product topology and the box topology?

On the Cartesian product of topological spaces, there are two standard topologies: One is the product topology, the other is the box topology.

As is well known, the box topology is generated by the product of open sets, and the product topology is generated by such products with the restriction that only finitely many factors are not the full space.

However in principle there could be other topologies defined on the product which sit “in between” product and box topology. As a simple example, one might demand that at most *countably* many factors are not the full space. Or one might take the index set to have a topology, and demand that the set of indices corresponding to non-full spaces is relatively compact.

My question now is: Are there any such ”in between” topologies that are actually known to be useful?

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