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 [math]\displaystyle{ \mathcal{F} }[/math] is of finite character provided it has the following properties:
- For each [math]\displaystyle{ A\in \mathcal{F} }[/math], every finite subset of [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math].
- If every finite subset of a given set [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math], then [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math].
Statement of the lemma
Let [math]\displaystyle{ Z }[/math] be a set and let [math]\displaystyle{ \mathcal{F}\subseteq\mathcal{P}(Z) }[/math]. If [math]\displaystyle{ \mathcal{F} }[/math] is of finite character and [math]\displaystyle{ X\in\mathcal{F} }[/math], then there is a maximal [math]\displaystyle{ Y\in\mathcal{F} }[/math] (according to the inclusion relation) such that [math]\displaystyle{ X\subseteq Y }[/math].[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 [math]\displaystyle{ \mathcal{F} }[/math] 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). 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]
 |  |
