Following is the definition of Infinite Cartesian Products.
For cartesian product, why do not we just use cartesian product and get "n-tuples" instead of "arbitrary(possibly infinite) indexed family" of function sets?
I am not sure but the reason might be that there is not any usual way to select from tuples unless a function used so functions used in first place. (Actually there is projection map definition in wiki article but not sure about it.) (This answer is also about n-tuple item access; Mathematical symbol to reference the i-th item in a tuple?)
So, is not there any way other than functions to access n-tuple items? So, what is the point actually?