Let A be a finite set of topological spaces $\ X_\alpha $ set. Now let us consider the product set of this topological spaces P=$\ \prod_\alpha X_\alpha $.
Now a topology in P , with all sets of the form $\ \prod_\alpha U_\alpha $, where U is open in $ X_\alpha $ as its basis is called a box topology and a topology in P ,with all sets of the form $\ \prod_\alpha U_\alpha $, with $\ U_\alpha $ = $\ X_\alpha $ for all but for finitely many $\ \alpha $ $\ \epsilon $ K,where K is some indexing set is called a product topology.
What is the significance of the later mentioned finitely many sets in the definition of the product topology?
The lecture notes I am going through mention that box topology is too coarse to be useful, it generates a lots of open sets, and as such product topology is more preferable. Can I have a rigorous explanation for this?