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 is of finite character provided it has the following properties:

1. For each , every finite subset of belongs to .
2. If every finite subset of a given set belongs to , then belongs to .

Statement of the lemma

Let be a set and let . If is of finite character and , then there is a maximal (according to the inclusion relation) such that .

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 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

References

• Brillinger, David R. "John Wilder Tukey" http://www.ams.org/notices/200202/fea-tukey.pdf