# Why do we need the axiom of choice in showing the non-emptiness of an infinite Cartesian product

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!

• "pick arbitrarly" <- That's choice. In fact, the axiom of choice is precisely the assertion that a product of nonempty sets always is nonempty. – Daniel Fischer Jul 20 '14 at 13:18

Picking an arbitrary $x_i$ is exactly where the axiom of choice gets into the picture.
Otherwise, how can you justify the existence of such a function which picks this arbitrary $x_i$?
Recall that the axiom of choice is needed when the product is infinite. When the product is finite, then indeed by induction we can choose an arbitrary $x_i$ and it's fine. But this induction is not transfinite.