Teichmüller–Tukey lemma


In mathematics, the Teichmüller–Tukey 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 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.