teichmüller, tukey, lemma, mathematics, sometimes, named, just, tukey, lemma, named, after, john, tukey, oswald, teichmüller, lemma, that, states, that, every, nonempty, collection, finite, character, maximal, element, with, respect, inclusion, over, zermelo, . In mathematics the Teichmuller Tukey lemma sometimes named just Tukey s lemma named after John Tukey and Oswald Teichmuller 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 Teichmuller 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 Contents 1 Definitions 2 Statement of the lemma 3 Applications 4 Notes 5 ReferencesDefinitions EditA family of sets F displaystyle mathcal F is of finite character provided it has the following properties For each A F displaystyle A in mathcal F every finite subset of A displaystyle A belongs to F displaystyle mathcal F If every finite subset of a given set A displaystyle A belongs to F displaystyle mathcal F then A displaystyle A belongs to F displaystyle mathcal F Statement of the lemma EditLet Z displaystyle Z be a set and let F P Z displaystyle mathcal F subseteq mathcal P Z If F displaystyle mathcal F is of finite character and X F displaystyle X in mathcal F then there is a maximal Y F displaystyle Y in mathcal F according to the inclusion relation such that X Y displaystyle X subseteq Y 2 Applications EditIn linear algebra the lemma may be used to show the existence of a basis Let V be a vector space Consider the collection F displaystyle 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 Notes Edit Jech Thomas J 2008 1973 The Axiom of Choice Dover Publications ISBN 978 0 486 46624 8 Kunen Kenneth 2009 The Foundations of Mathematics College Publications ISBN 978 1 904987 14 7 References EditBrillinger David R John Wilder Tukey 1 Retrieved from https en wikipedia org w index php title Teichmuller Tukey lemma amp oldid 1106927375, wikipedia, wiki, book, books, library,