Comparing box topology and product topology, there are many better theorems standing in product topology, but not stands in box topology.
So is it useful to study box topology?
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Probably not. The box topology doesn't behave well enough and it doesn't appear often enough for us to care. Actually I don't recall a single useful result based on box topology except for "counterexamples in topology" style results. – freakish Jan 07 '21 at 08:23
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1Box products and modifications of box products have a number of counterexamples in topology, including the first example in $\mathsf{ZFC}$ of a Dowker space. And counterexamples to plausible conjectures are important. – Brian M. Scott Jan 07 '21 at 08:36
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In the list of related questions there is a question Is the box topology good for anything? with 62 upvotes and 5 answers. – Alex Ravsky Jan 07 '21 at 20:36