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.

Definitions
A family of sets $$\mathcal{F}$$ is of finite character provided it has the following properties:
 * 1) For each $$A\in \mathcal{F}$$, every finite subset of $$A$$ belongs to $$\mathcal{F}$$.
 * 2) If every finite subset of a given set $$A$$ belongs to $$\mathcal{F}$$, then $$A$$ belongs to  $$\mathcal{F}$$.

Statement of the lemma
Let $$Z$$ be a set and let $$\mathcal{F}\subseteq\mathcal{P}(Z)$$. If $$\mathcal{F}$$ is of finite character and $$X\in\mathcal{F}$$, then there is a maximal $$Y\in\mathcal{F}$$ (according to the inclusion relation) such that $$X\subseteq Y$$.

Applications
In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider the collection $$\mathcal{F}$$ of linearly independent sets of vectors. This is a collection of finite character. Thus, a maximal set exists, which must then span V and be a basis for V.