Hausdorff maximality theorem
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The Hausdorff maximality theorem, (also called the Hausdorff maximal principle) formulated and proved by Felix Hausdorff in 1914, is an alternate 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.