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 $A\in \mathcal{F}$, 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

Whenever $\mathcal{F}\subseteq\mathcal{P}(A)$ is of finite character and $X\in\mathcal{F}$, there is a maximal $Y\in\mathcal{F}$ such that $X\subseteq Y$.^[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 $\mathcal{F}$ 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"