# 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.