maximality principle

Let $S$ be a collection of sets. If, for each chain $C\subseteq S$, there exists an $X\in S$ such that every element of $C$ is a subset of $X$, then $S$ contains a maximal element. This is known as the maximality principle.

The maximality principle is equivalent to the axiom of choice.

Title	maximality principle
Canonical name	MaximalityPrinciple
Date of creation	2013-03-22 12:26:18
Last modified on	2013-03-22 12:26:18
Owner	akrowne (2)
Last modified by	akrowne (2)
Numerical id	9
Author	akrowne (2)
Entry type	Theorem
Classification	msc 03E30
Classification	msc 03E25
Synonym	maximal principle
Related topic	ZornsLemma
Related topic	AxiomOfChoice
Related topic	WellOrderingPrinciple
Related topic	TukeysLemma
Related topic	ZermelosPostulate
Related topic	HaudorffsMaximumPrinciple