Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 6 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.