Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am not an an expert in set theory, so this question could be trivial. I am sorry in that case.

Let $I$ be a set and $\{ X_i \}_{i \in I}$ be a collection of sets such that $X_i \neq \emptyset$ for all $i \in I$. The axiom of choice tells precisely that the set $$ \prod_{i \in I} X_i \neq \emptyset $$ is not empty. The use of the word "choice" here is clear. To produce an element $(x_i)\in \prod_{i \in I} X_i$ we need to choose an $x_i \in X_i$ for all $i \in I$.

Now I am wondering: what if there no choice at all? Namely, if $X_i = \{x_i\}$ has one element for each $i \in I$? Indeed, in this case $\prod_{i \in I} X_i$ has only one element, that is $(x_i)$, and there is no choice at all. So the question is the following:

Question: it the axiom of choice needed to prove that the product of a family of sets, each with exactly one element, is not empty?

Thanks!

share|improve this question
    
Of course not. Please read the FAQ mathoverflow.net/faq –  Martin Brandenburg Apr 23 '13 at 12:58
1  
No, obviously not. It is also not needed to prove that products of finite families of nonempty sets are nonempty, but this is a bit more subtle. In your case, you can just explicitly define the function, and the basic axioms of set theory suffice to verify that this definition gives you a set. –  Andres Caicedo Apr 23 '13 at 12:59
1  
The questino is probably not appropriate for this site. No, AC is not needed to choose from a family of one-element sets. It may be needed, however, to choose from a family of two-element sets. –  Gerald Edgar Apr 23 '13 at 13:11
1  
Note that if $X_i=\lbrace x_i\rbrace$, then $\bigcup_{i\in I}\lbrace\langle i,x_i\rangle\rbrace$ is a choice function. –  Asaf Karagila Apr 23 '13 at 13:20
1  
Let $U$ be the union of the sets $X_i$. Let $Y=I\times U$. For $(i,x)\in Y$, let $\phi(i,x)$ be the property $x\in X_i$. Then the axiom schema of specification gives us a set $\{(i,x)|i\in I,x\in X_i\}$, which is the product set you're looking for. –  Steven Landsburg Apr 23 '13 at 13:20
show 1 more comment

closed as too localized by Martin Brandenburg, Henry Cohn, Misha, Gerald Edgar, Asaf Karagila Apr 23 '13 at 13:18

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.