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.^[1]

Definitions

A family of sets ${\mathcal {F}}$ is of finite character provided it has the following properties:

For each $A\in {\mathcal {F}}$ , every finite subset of $A$ belongs to ${\mathcal {F}}$ .
If every finite subset of a given set $A$ belongs to ${\mathcal {F}}$ , then $A$ belongs to ${\mathcal {F}}$ .

Statement of the lemma

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

^ Jech, Thomas J. (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 978-0-486-46624-8.
^ Kunen, Kenneth (2009). The Foundations of Mathematics. College Publications. ISBN 978-1-904987-14-7.

References

Brillinger, David R. "John Wilder Tukey" [1]

[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.

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