Hausdorff’s maximum principle
Theorem
Let be a partially ordered set. Then there exists a maximal totally
ordered
subset of .
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 be the set of all totally ordered subsets of . is not empty, since the empty set is an element of . Partial order
by inclusion. Let be a chain (of elements) in . Being each totally ordered, the union of all these elements of is again a totally ordered subset of , and hence an element of , as is easily verified. This shows that , 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 |