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# Zorns lemma

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Let P be a partially ordered set. Zorn's Lemma states that if every totally ordered subset of P has an upper bound in P, then P has a (not necessarily unique) maximal element. Here, an "upper bound" is an element in P which is at least as big as every element from the totally ordered subset; it does not have to be a member of that subset. A "maximal element" of P is an element so that no other element of P is bigger.

Like the well-ordering principle, Zorn's Lemma is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms (see set theory) is sufficient to prove the other. It is probably the most useful of all equivalents of the axiom of choice and occurs in the proofs of several theorems of crucial importance, for instance the [Hahn-Banach theorem]? in [functional analysis]?, the theorem that every vector space has a basis, and the theorem that every field has an [algebraic closure]?.

Zorn's Lemma is named after Max Zorn.

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Last edited August 16, 2001 2:50 pm (diff)
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