Hausdorff maximal principle

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The Hausdorff maximal principle, (also called the Hausdorff maximality theorem) formulated and proved by Felix Hausdorff in 1914, is an alternate and earlier formulation of Zorn's lemma and therefore also equivalent to the axiom of choice.

It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset (i.e. in a totally ordered subset which, if enlarged in any way, does not remain totally ordered: in general, there are many maximal totally ordered subsets containing a given totally ordered subset).

An equivalent (but not obviously so) form of the theorem is that in every partially ordered set there exists a maximal totally ordered subset.

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