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Let $A$ and $B$ be uncountable sets; with arbitrary elements $a\in A$ and $b\in B$. Further, let $f:A\to B$ me a map. I have two-related questions:

  1. Can I define the cartesian product $\prod_{a\in A}f(a)$?

  2. If the above is yes, is is true that $\prod_{a\in A}f(a)\equiv\prod_{n=1}^{\infty}f(a)$?

I have doubts in (1) because $A$ is uncountable, and therefore I’m not sure whether the expression “$\prod_{a\in A}$” makes any sense —unless $\prod_{a\in A}$ is to be read as $\prod_{n=1}^{\infty}$. Hence, question (2).

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Well, yes. Given a function $F:A\to B$, its cartesian product is $$\prod_{a\in A}F(a)=\left\{g\,:\, \left(g\text{ is a function }A\to \cup_{a\in A}f(a) \right)\land \forall a\in A, g(a)\in f(a)\right\}$$

There is no restriction whatsoever on $A$ (which could even be empty) or $B$ (which is kind of irrelevant, if you have the axiom of replacement).

When $A=\Bbb N$, you may find the notation $\prod_{n=0}^\infty F(n)$ as an alternative to $\prod_{n\in\Bbb N} F(n)$, especially in situations where you want to make a simplification such as $\prod_{n=m}^\infty F(n):=\prod_{n\in\{k\in\Bbb N\,:\, k\ge m\}} F(n)$. Personally, I think that $\prod_{n\in\Bbb N}$ should be preferred unless it serves such a purpose.

It should be noted that in many textbooks the indexed-family notation $\{F_a\}_{a\in A}$ may be preferred over the function notation, though technically they indicate the same object.

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  • $\begingroup$ You seem to assume each $F(a)$ is some set? $\endgroup$ Commented Nov 3, 2021 at 9:31
  • $\begingroup$ @HennoBrandsma well, yes. $\endgroup$ Commented Nov 3, 2021 at 13:54
  • $\begingroup$ maybe make clear to the OP that that is what you’re assuming or the product would make no sense $\endgroup$ Commented Nov 3, 2021 at 13:55
  • $\begingroup$ @HennoBrandsma If it isn't an implicit assumption of the question already, then I'd much rather remove the answer then either address it or give a partial one. $\endgroup$ Commented Nov 3, 2021 at 13:57
  • $\begingroup$ There is no notion of the Cartesian product of a function. Only for the special case of an indexed family of sets. $\endgroup$ Commented Nov 3, 2021 at 14:00

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