Is an order necessary in a finite or infinite cartesian product? The elements of $\displaystyle \prod_{i\in I}X_i$ are usually represented by tuples. For this cartesian product to be defined, is it necessary for $I$ to be an ordered set?

I guess the answer is no because by definition, the product is just the set of all functions $f: I \to \bigcup_{i \in I} X_i$ such that $f(i)\in X_i$ for each $i\in I.$
Remark: how do we know that this product is not empty? If $X$ is a set, then the Axiom of Choice says that for each $x\in X$,  there is a function $f$ that maps $x$ into $\bigcup X$, such that $f(x)\in x.$ Now, if $X=\{X_i\}_{i\in I}$ this means that for each $X_i\in X,$ there is a function $f$ that sends $X_i$ to an element of $X_i$, and therefore, the composition $f\circ g:I\to \bigcup_{i \in I} X_i$, satisfies the definition of the product, where $g(i)=X_i.$ That is, the product is not empty.
