Zorn's lemma

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

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Every partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.

Related Topics:
Partially ordered set - Chain - Totally ordered - Subset - Upper bound - Maximal element

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It is named after the mathematician Max Zorn.

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The terms are defined as follows. Suppose (P,≤) is the partially ordered set. A subset T is totally ordered if for any s, t in T we have either s ≤ t or t ≤ s. Such a set T has an upper bound u in P if t ≤ u 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 the only element x in P with m ≤ x is x = m itself.

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Like the well-ordering theorem, Zorn's lemma is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms of set theory is sufficient to prove the other. 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 ring has a maximal ideal and that every field has an algebraic closure.

Related Topics:
Well-ordering theorem - Axiom of choice - Zermelo-Fraenkel axioms - Set theory - Hahn-Banach theorem - Functional analysis - Vector space - Basis - Tychonoff's theorem - Topology - Compact spaces - Abstract algebra - Ring - Maximal ideal - Field - Algebraic closure

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