Hausdorff maximality theorem
The
Hausdorff maximality theorem, 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.