Hausdorff maximal principle

Hausdorff maximal principle
Tag	`0KM4KVR9MV`
URI	`m2://0KM4KVR9MV`
Aliases	`maximal-chain-principle`
Kind	statement, theorem
Writing	W1 stated
MSC	03E25
Status	This copy of the site has no build data. Run `m2 status 0KM4KVR9MV`.
Uses	Chain, Maximal element, Axiom of choice
Same as	Wikidata Q738328 Wikipedia Hausdorff maximal principle

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Statement

Assume the axiom of choice. Let $(P, \le)$ be a partially ordered set. Then each chain of $P$ is a subset of a maximal chain. A maximal chain is a chain that is not a proper subset of another chain of $P$ .

Remarks

Over ZF, the principle is equivalent to the axiom of choice and to Zorn's lemma. With the empty chain, it says that each partially ordered set has a maximal chain. Felix Hausdorff proved it in 1914.

Explanations

L1 human

Start with any chain in the order. You can extend it until no element can be added without breaking the chain property.

What this node needs for W2

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Sources

Web https://en.wikipedia.org/wiki/Hausdorff_maximal_principle, Statement

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