Axiom of Choice

Statement

To any nonempty family $X$ whose elements are nonempty sets, there exists a function called a choice function

such that $f(X_i) \in X_i$ for all $X_i \in X$

Equivalent Statement

Order by alphabet

1. (HMP) The Hausdorff Maximality Principle
2. (WOP) The Well-Ordering Principle
3. (ZL) Zorn’s Lemma

Prove

1. AC to HMP
2. HMP to ZL
3. ZL to WOP
4. WOP to AC