Hausdorff maximal principle

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over Zermelo–Fraenkel set theory. The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33).

Statement

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. Here a maximal totally ordered subset is one that, if enlarged in any way, does not remain totally ordered. The maximal set produced by the principle is not unique, in general; there may be many maximal totally ordered subsets containing a given totally ordered subset.

An equivalent form of the principle is that in every partially ordered set there exists a maximal totally ordered subset.

To prove that it follows from the original form, let A be a poset. Then $\varnothing$ is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing $\varnothing$ , in particular A contains a maximal totally ordered subset.

For the converse direction, let A be a partially ordered set and T a totally ordered subset of A. Then

$\{S\mid T\subseteq S\subseteq A{\mbox{ and S totally ordered}}\}$ is partially ordered by set inclusion $\subseteq$ , therefore it contains a maximal totally ordered subset P. Then the set $M=\bigcup P$ satisfies the desired properties.

The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is very similar to this proof.