Teichmüller–Tukey lemma

A maximality principle stating that every family of finite character has a maximal member under inclusion. It is equivalent to the axiom of choice.

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Finite character: A property of sets determined by their finite subsets: a set has the property exactly when all its finite subsets do. This condition is the lemma’s key hypothesis, ensuring membership can be checked through finite subsets.
Set inclusion: The relation in which every element of one set also belongs to another. The lemma orders candidate members by inclusion, rather than by size or another measure.
Hamel basis: A basis of a vector space in which every vector is a finite linear combination of basis elements. A linearly independent set has finite character, so the lemma extends it to a maximal independent set.
Andreas Speiser: A Swiss mathematician whose work included an early formulation of the maximality principle now associated with Teichmüller and Tukey. His earlier statement is part of the lemma’s historical lineage.
Partially ordered set: A set equipped with a reflexive, antisymmetric, and transitive relation. The lemma seeks a maximal family under the partial order of inclusion.
Axiom of choice: The principle that every family of nonempty sets has a choice function. The lemma is equivalent to this foundational principle, despite its different maximality language.
Maximal ideal: A proper ideal contained in no larger proper ideal of a ring. The lemma can extend a proper ideal to a maximal one because properness is preserved by chain unions.
Oswald Teichmüller: A German mathematician known for work in complex analysis and the theory of quasiconformal mappings. His 1939 work gave a formulation that contributed to the lemma’s name.
Maximal element: An element of a partially ordered set with no strictly greater element. The lemma guarantees this endpoint, which need not be a greatest element.
Well-ordering theorem: The theorem that every set can be given a well-order. It is another statement equivalent to the axiom of choice and helps locate the lemma among choice principles.