Editing Hausdorff maximal principle (section)

{{Short description|Mathematical result or axiom on order relations}}

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 [[partial order|partially ordered set]], every [[total order|totally ordered]] [[subset]] is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion.

In a partially ordered set, a totally ordered subset is also called a chain. Thus, the maximal principle says every chain in the set extends to a maximal chain.

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).