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How can you proof that the Teichmüller-Tukey Lemma (which says that if $S$ is nonempty and of finite character, $S$ contains a maximal element with respect to the subset ordering), implies the Axiom of Choice?

Any hints or solutions will be appreciated!

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The proof is not difficult, let $\{A_i\mid i\in I\}$ be a family of non-empty sets. Now consider $\scr F$ to be the collection of all partial choice functions, that is every choice function on a subset of $I$.

First note that any maximal element must be a full choice function on $I$, otherwise we can always extend it by one more element. So if we can prove $\scr F$ has a finite character we're essentially done.

But now, suppose that $f$ is a partial choice function, then every subset of $f$ is a partial choice function, in particular its finite subsets. And if $A$ is a set such that every finite subset of $A$ is in $\scr F$ then $A$ is a partial choice function, and therefore in $\scr F$. And therefore we're done.

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