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.^[1]

Definitions

A family of sets ${\displaystyle {\mathcal {F}}}$ is of finite character provided it has the following properties:

1. For each ${\displaystyle A\in {\mathcal {F}}}$, every finite subset of ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.
2. If every finite subset of a given set ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$, then ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.

Statement of the lemma

Let ${\displaystyle Z}$ be a set and let ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {P}}(Z)}$. If ${\displaystyle {\mathcal {F}}}$ is of finite character and ${\displaystyle X\in {\mathcal {F}}}$, then there is a maximal ${\displaystyle Y\in {\mathcal {F}}}$ (according to the inclusion relation) such that ${\displaystyle X\subseteq Y}$.^[2]

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 ${\displaystyle {\mathcal {F}}}$ 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.