Zorn lemma

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

maximal principle

If in a non-empty partially ordered set $X$ every totally ordered subset (cf. Totally ordered set) has an upper bound, then $X$ contains a maximal element. An element $x_0$ is called an upper bound of a subset $A\subset X$ if $x\leq x_0$ for all $x\in A$. If an upper bound for $A$ exists, then the set $A$ is said to be bounded above. An element $x_0\in X$ is called maximal in $X$ if there is no element $x\in X$, $x\not=x_0$, such that $x_0\leq x$.

The lemma was stated and proved by M. Zorn in [Zo]. It is equivalent to the axiom of choice.

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Earlier versions of the maximal principle, differing in detail from the one stated above but logically equivalent to it, were introduced independently by several mathematicians, the earliest being F. Hausdorff in 1909. For accounts of the history of the maximal principle, see [Ca]–[RuRu].

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