I need to show that Hausdorff Maximal Principle is equivalent to the Axiom of Choice. Suggested is to use Tukeys Lemma.
So far I have that Hausdorff Maximal Principle states that whenever < is a strict partial order of a set A, there is a maximal chain C $\subseteq$ A.
Tukeys Lemma states that $F \subseteq P(A)$ is of finite character if and only if for all $X \subseteq A : X \in F$ iff every finite subset of X is in $F$.
This is coming from my book Kunen Foundations of Mathematics.
I also understand a chain to be a totally ordered set, if for all x,y, in X, either $x \leq y$ or $y \leq x$. A chain has at most one maximal element.
I am really stuck on how to relate these two to each other. I think I have an idea of how to show equivalence from Tukey's Lemma to Axiom of Choice. But I am stuck when it comes to showing that the Hausdorff Maximal Principle is equivalent to Tukeys Lemma. Any help or references is greatly appreciated. I have also been reading up on this website: http://web.science.mq.edu.au/~chris/sets/CHAP09%20Axiom%20of%20Choice.pdf