# 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 ZF (Zermelo–Fraenkel set theory without the axiom of choice); the principle is also called the **Hausdorff maximality theorem** or the **Kuratowski lemma** (Kelley 1955:33).

## Statement[edit]

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.

## Examples[edit]

**EXAMPLE 1.** If *A* is any collection of sets, the relation "is a proper subset of" is a strict partial order on *A*. Suppose that *A* is the collection of all circular regions (interiors of circles) in the plane. One maximal totally ordered sub-collection of *A* consists of all circular regions with centers at the origin. Another maximal totally ordered sub-collection consists of all circular regions bounded by circles tangent from the right to the y-axis at the origin.

**EXAMPLE 2.** If (x_{0}, y_{0}) and (x_{1}, y_{1}) are two points of the plane ℝ^{2}, define (x_{0}, y_{0}) < (x_{1}, y_{1})

if y_{0} = y_{1} and x_{0} < x_{1}. This is a partial ordering of ℝ^{2} under which two points are comparable only if they lie on the same horizontal line; the maximal totally ordered sets are horizontal lines in ℝ^{2}.

## References[edit]

- John Kelley (1955),
*General topology*, Von Nostrand. - Gregory Moore (1982),
*Zermelo's axiom of choice*, Springer. - James Munkres (2000),
*Topology*, Pearson.