Following is the definition of Infinite Cartesian Products.

enter image description here https://en.wikipedia.org/wiki/Cartesian_product

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?

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    $\begingroup$ What exactly is an "$n$-tuple" when $n$ is infinite? The point is that we actually want to think of ordinary finite tuples themselves as functions: a $2$-tuple, for example, is a function with domain $\{0,1\}$, whose value on $0$ is the $0$th coordinate and whose value on $1$ is the $1$st coordinate. $\endgroup$ – Noah Schweber Oct 6 at 22:21
  • $\begingroup$ So, is it just because of that we cannot define an infinite n-tuple? $\endgroup$ – lockedscope Oct 6 at 22:44
  • $\begingroup$ I wouldn't phrase it that way. Rather, I'd say that we haven't actually defined finite tuples in a sufficiently precise way. Try to define (say) a $3$-tuple in a precise way and I think you'll see the issue. $\endgroup$ – Noah Schweber Oct 6 at 22:47
  • $\begingroup$ @NoahSchweber are not tuples necessarily ordered? So, i understand like this, they are distinct by order but when we are building/defining them, they do not have any order. $\endgroup$ – lockedscope Oct 6 at 22:47
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    $\begingroup$ @lockedscope: There was a user that insisted that we can prove the axiom of choice because we can apply a Python/JS like approach and if we have a collection of sets, we can simply x.map((y)=>some element of y), "programming logic" can be misleading if you don't really think about the underlying math below the actual operations. Once you realise that the iteration is really just hiding a for loop, you can see that there will be some indexing and finite-to-infinite failure (because in a proof for or while get unfolded to their linear execution, in some sense). The same goes here. $\endgroup$ – Asaf Karagila Oct 6 at 23:33