Hausdorff’s maximum principle

Theorem Let $X$ be a partially ordered set. Then there exists a maximal totally ordered subset of $X$.

The Hausdorff’s maximum principle is one of the many theorems equivalent to the axiom of choice (http://planetmath.org/AxiomOfChoice). The below proof uses Zorn’s lemma, which is also equivalent to the .

Proof.

Let $S$ be the set of all totally ordered subsets of $X$. $S$ is not empty, since the empty set is an element of $S$. Partial order $S$ by inclusion. Let $\tau$ be a chain (of elements) in $S$. Being each totally ordered, the union of all these elements of $\tau$ is again a totally ordered subset of $X$, and hence an element of $S$, as is easily verified. This shows that $S$, ordered by inclusion, is inductive. The result now follows from Zorn’s lemma. ∎

Title	Hausdorff’s maximum principle
Canonical name	HausdorffsMaximumPrinciple
Date of creation	2013-03-22 13:04:42
Last modified on	2013-03-22 13:04:42
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 03E25
Synonym	maximum principle
Synonym	Hausdorff maximality theorem
Related topic	ZornsLemma
Related topic	AxiomOfChoice
Related topic	ZermelosWellOrderingTheorem
Related topic	ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple
Related topic	EveryVectorSpaceHasABasis
Related topic	MaximalityPrinciple