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'Hausdorff's maximum principle'
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| Title of object: |
Hausdorff's maximum principle |
| Canonical Name: |
HaudorffsMaximumPrinciple |
| Type: |
Theorem |
| Created on: |
2002-09-29 19:48:23 |
| Modified on: |
2004-02-17 00:03:29 |
| Classification: |
msc:03E25 |
| Synonyms: |
Hausdorff's maximum principle=maximum principle |
Revision comment (for changes between this and next version):
| Changes for correction #7280 ('bad link'). |
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Content:
{\bf 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.
{\bf 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 $S$.
Given a subset $\tau$ of $S$, the union of
all the elements of $\tau$ is again 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.$\Box$ |
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