The goal of this note is to show the following result:

Theorem 1The following statements are equivalent in

The axiom of choice:Every set can be well-ordered.- Every collection of nonempty set admits a choice function, i.e., if for all then there is such that for all
Zorn’s lemma:If is a partially ordered set with the property that every chain has an upper bound, then has maximal elements.- Any family of pairwise disjoint nonempty sets admits a
selector, i.e., a set such that for all in the family.- Any set is a well-ordered union of finite sets of bounded size, i.e., for every set there is a natural an ordinal and a function such that for all and
Tychonoff’s theorem:The topological product of compact spaces is compact.- Every vector space (over any field) admits a basis.