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Hausdorff's maximum principle
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(Theorem)
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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. The below proof uses Zorn's lemma, which is also equivalent to the axiom of choice.
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. 
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"Hausdorff's maximum principle" is owned by CWoo. [ full author list (3) | owner history (2) ]
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Cross-references: union, chain, inclusion, partial order, element, empty set, Zorn's lemma, proof, equivalent, subset, totally ordered, partially ordered set, theorem
There are 4 references to this entry.
This is version 9 of Hausdorff's maximum principle, born on 2002-09-29, modified 2008-03-25.
Object id is 3491, canonical name is HaudorffsMaximumPrinciple.
Accessed 8432 times total.
Classification:
| AMS MSC: | 03E25 (Mathematical logic and foundations :: Set theory :: Axiom of choice and related propositions) |
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Pending Errata and Addenda
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