Zorn’s lemma

If $X$ is a partially ordered set such that every chain in $X$ has an upper bound, then $X$ has a maximal element.

Note that the empty chain in $X$ has an upper bound in $X$ if and only if $X$ is non-empty. Because this case is rather different from the case of non-empty chains, Zorn’s Lemma is often stated in the following form: If $X$ is a non-empty partially ordered set such that every non-empty chain in $X$ has an upper bound, then $X$ has a maximal element. (In other words: Any non-empty inductively ordered set has a maximal element.)

In ZF, Zorn’s Lemma is equivalent to the Axiom of Choice (http://planetmath.org/AxiomOfChoice).

Title	Zorn’s lemma
Canonical name	ZornsLemma
Date of creation	2013-03-22 12:09:04
Last modified on	2013-03-22 12:09:04
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	10
Author	yark (2760)
Entry type	Theorem
Classification	msc 06A06
Classification	msc 03E25
Related topic	AxiomOfChoice
Related topic	MaximalityPrinciple
Related topic	HaudorffsMaximumPrinciple
Related topic	ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple
Related topic	EveryVectorSpaceHasABasis
Related topic	TukeysLemma
Related topic	ZermelosPostulate
Related topic	KuratowskisLemma
Related topic	EveryRingHasAMaximalIdeal
Related topic	InductivelyOr