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

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

  1. For each , every finite subset of belongs to .
  2. If every finite subset of a given set belongs to , then belongs to .

Statement of the lemma

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Let be a set and let . If is of finite character and , then there is a maximal (according to the inclusion relation) such that .[2]

Applications

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In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider the collection 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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  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.

References

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