Teichmüller–Tukey lemma

In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equivalent to the axiom of choice, and therefore to the well-ordering theorem, Zorn's lemma, and the Hausdorff maximal principle.

A family of sets is of finite character provided it has the following properties:

Whenever ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {P}}(A)}$ is of finite character and ${\displaystyle X\in {\mathcal {F}}}$, there is a maximal ${\displaystyle Y\in {\mathcal {F}}}$ such that ${\displaystyle X\subseteq Y}$.

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