# Axiom of choice and the empty set

Could someone explain to me why it is important for a set to be non-empty when working with a choice function?

• Because you can't choose a member of the empty set. ${}\qquad{}$ – Michael Hardy Nov 24 '15 at 19:45

Because a choice function, working on $\emptyset$, would be a function for which $f(\emptyset)\in\emptyset$, which is obviously untrue.
Well, this depends on your definition of a choice function for a set $A$. I use the following definition, which I shall use in this answer:
A choice function for a set $A$ is a function $f:\mathcal P \left({A}\right)-\{\emptyset\} \rightarrow A$ such that $f(B) \in B$ for every non-empty subset $B$ of $A$.