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Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:
It is named after the mathematicians Max Zorn and Kazimierz Kuratowski. ==Background== The terms used in the statement of the lemma 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 ''s'' ≤ ''t'' or ''t'' ≤ ''s''. Any such totally ordered set ''T'' is called a ''chain''. 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''. An element ''m'' of ''P'' is called a ''maximal element'' (or ''non-dominated'') if there is no element ''x'' in ''P'' for which ''m'' < ''x''. Note that ''P'' is not explicitly required to be non-empty. However, the empty set is a chain (trivially), hence is required to have an upper bound, thus exhibiting at least one element of ''P''. An equivalent formulation of the lemma is therefore:
The distinction may seem subtle, but proofs involving Zorn's lemma often involve taking a union of some sort to produce an upper bound. The case of an empty chain, hence empty union is a boundary case that is easily overlooked. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zorn's lemma」の詳細全文を読む スポンサード リンク
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