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 $\mathcal$ is of finite character provided it has the following properties:
#For each $A\in \mathcal$, every finite subset of $A$ belongs to $\mathcal$.
#If every finite subset of a given set $A$ belongs to $\mathcal$, then $A$ belongs to $\mathcal$.

Statement of the lemma

Let $Z$ be a set and let $\mathcal\subseteq\mathcal(Z)$. If $\mathcal$ is of finite character and $X\in\mathcal$, then there is a maximal $Y\in\mathcal$ (according to the inclusion relation) such that $X\subseteq Y$.

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 $\mathcal$ 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


{{DEFAULTSORT:Teichmuller-Tukey lemma
Families of sets
Order theory
Axiom of choice
Lemmas in set theory