502 – Equivalents of the axiom of choice

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

Theorem 1 The following statements are equivalent in

 

1. The axiom of choice: Every set can be well-ordered.
2. Every collection of nonempty set admits a choice function, i.e., if for all then there is such that for all
3. Zorn’s lemma: If is a partially ordered set with the property that every chain has an upper bound, then has maximal elements.
4. Any family of pairwise disjoint nonempty sets admits a selector, i.e., a set such that for all in the family.
5. 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
6. Tychonoff’s theorem: The topological product of compact spaces is compact.
7. Every vector space (over any field) admits a basis.