# 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 is a totally ordered subset of *A*, hence there exists a maximal totally ordered subset containing , 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 , therefore it contains a maximal totally ordered subset *P*. Then the set satisfies the desired properties.

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

## References

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