Hausdorff’s maximum principle

Theorem Let X be a partially ordered setMathworldPlanetmath. Then there exists a maximal totally orderedPlanetmathPlanetmath subset of X.

The HausdorffPlanetmathPlanetmath’s maximum principle is one of the many theorems equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the axiom of choiceMathworldPlanetmath ( The below proof uses Zorn’s lemma, which is also equivalent to the .


Let S be the set of all totally ordered subsets of X. S is not empty, since the empty setMathworldPlanetmath is an element of S. Partial orderMathworldPlanetmath S by inclusion. Let τ be a chain (of elements) in S. Being each totally ordered, the union of all these elements of τ 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