# Teichmüller–Tukey lemma

In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, 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 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

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 ${\displaystyle X\subseteq Y}$.[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.

## Notes

1. ^ Jech, Thomas J. (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 978-0-486-46624-8.
2. ^ Kunen, Kenneth (2009). The Foundations of Mathematics. College Publications. ISBN 978-1-904987-14-7.

## References

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