Zorn's lemma

 related topics {math, number, function}

Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:

Suppose a partially ordered set has the property that every chain (i.e. totally ordered subset) has an upper bound. Then the set contains at least one maximal element.

It is named after the mathematicians Max Zorn and Kazimierz Kuratowski.

The terms are defined as follows. Suppose (P,≤) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have st or ts. Such a set T has an upper bound u in P if tu for all t in T. Note that u is an element of P but need not be an element of T. A maximal element of P is an element m in P such that for no element x in P, m < x.

Zorn's lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the others. It 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, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

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An example application

We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring R with unity contains a maximal ideal. The set P here consists of all (two-sided) ideals in R except R itself, which is not empty since it contains at least the trivial ideal {0}. This set is partially ordered by set inclusion. We are done if we can find a maximal element in P. The ideal R was excluded because maximal ideals by definition are not equal to R.