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.

References

• Brillinger, David R. "John Wilder Tukey" [1]