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]
A family of sets
l{F}
A\inl{F}
A
l{F}
A
l{F}
A
l{F}
Let
Z
l{F}\subseteql{P}(Z)
l{F}
X\inl{F}
Y\inl{F}
X\subseteqY
In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider the collection
l{F}