Given $I$ a set of indexes and $X_i$ a set of topological spaces, define
The Cartesian product: $\prod_{i \in I}X_i = \{ f:I \rightarrow \bigcup X_i | f(i) \in X_i \}$
I have read that we need the axiom of choice in order to show that the cartesian product of non-empty collection of non-empty sets is not empty.
My question is, why do we need the axiom of choice for that?
If all the spaces are non-empty, pick arbitrarly from each $X_i$ a point $x_i$, and the product is not empty.
I know that this what the axiom of choice is all about. But, The axiom states that we will have a well defined choice function. I don't see where we need a well defined choice function in the case of showing the non-emptyness of an infinite product..
Any help?
Thank you!