Teichmüller–Tukey lemma

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In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma, also known in certain circles as Tuki-Tuki lemma), named after John Tukey and Oswald Teichmüller, states that every nonempty collection of finite character has a maximal element with respect to inclusion. It is equivalent to the Axiom of Choice.

Definitions[edit]

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}.

Aplications[edit]

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.

See also[edit]

References[edit]

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