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 formula_1 is of finite character provided it has the following properties:
Statement of the lemma.
Let formula_9 be a set and let formula_10. If formula_1 is of finite character and formula_12, then there is a maximal formula_13 (according to the inclusion relation) such that formula_14.
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 formula_1 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.
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