Teichmüller–Tukey lemma

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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.[1]


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

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

Statement of the Lemma[edit]

Whenever \mathcal{F}\subseteq\mathcal{P}(A) is of finite character and X\in\mathcal{F}, there is a maximal Y\in\mathcal{F} such that X\subseteq Y.[2]


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.


  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. 


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