Hausdorff maximal principle
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In mathematics, 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.
[edit] Formal statement
The theorem can be stated in more formal terms. Let E be the collection of totally ordered subsets of a partially ordered set A. This collection E can itself be turned into a partially ordered system, simply by means of set inclusion, since the elements of E are subsets of A. That is, if 
, then 
if x is a subset of y, and so 
is a partially ordered system. The Hausdorff maximality theorem then states that E has a maximal element.
One possible proof of the theorem follows from Zorn's lemma, by noting that the union of subsets of E is again a subset of E, thus making E into a join-semilattice.
