# Teichmüller–Tukey lemma

(Redirected from Tukey's 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. It is equivalent to the Axiom of Choice.

## Definitions

A family of sets 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 $\mathcal{F}$.

## 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.