The Axiom of Choice reads:
The product of a collection of non-empty sets is non-empty.
As you know well, this axiom is equivalent to many other statements. A few examples (probably the most known) are the following:
- Zorn's lemma: Let $(P,\ge)$ be a poset in which every chain has an upper bound. Then $P$ has a maximal element.
This is used in a number of different places to prove very powerful theorems. A couple of examples I can think of right now are: the proof that every ring has a maximal ideal (commutative algebra) and the Hahn-Banach theorem (functional analysis).
- Well ordering theorem: Every set can be well-ordered (i.e. it admits a total order such that every subset has a least element).
- Tychonoff's theorem: Every product of compact topological spaces is compact.
This is used for example in the proof of Alaoglu's theorem (again functional analysis).
Wikipedia gives various more. Among them:
- Tarski's theorem: If $A$ is an infinite set, then there is a bijection $A\to A\times A$.
(As @Wicht dutifully commented, Tarski's theorem is the name given to the implication $|A|=A\times A \Rightarrow$ Axiom of Choice, while the other implication was known before the proof of this fact by Tarski.)
- Every vector space has a basis.
My question is: what are other useful statements which are equivalent to the axiom of choice? Where are they used, and to prove what?
P.s.: By "useful statements" I mean either statements which are important theorems by themselves (as Tychonoff's theorem) or that are directly used in (simple versions of) the proof of important theorems (as Zorn's lemma).