# Hausdorff maximal principle

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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 Zermeloâ€“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

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.

### Reference

• John Kelley (1955), General topology, Von Nostrand.
• Gregory Moore (1982), Zermelo's axiom of choice, Springer.
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