Teichmüller–Tukey lemma - Alchetron, the free social encyclopedia

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.

Definitions

A family of sets is of finite character provided it has the following properties:

For each A ∈ F , every finite subset of A belongs to F .
If every finite subset of a given set A belongs to F , then A belongs to F .

Statement of the Lemma

Whenever F ⊆ P ( A ) is of finite character and X ∈ F , there is a maximal Y ∈ F such that X ⊆ 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 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.

References

Teichmüller–Tukey lemma Wikipedia

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Contents

Definitions

Statement of the Lemma

Applications

References