Back Hausdorffs Maximalkettensatz German Principio maximal de Hausdorff Spanish Principe de maximalité de Hausdorff French עקרון המקסימום של האוסדורף HE Hausdorffov princip maksimalnosti Croatian Hausdorff–Birkhoff-tétel Hungarian Principio di massimalità di Hausdorff Italian Maximaal-principe van Hausdorff Dutch Twierdzenie Hausdorffa o łańcuchu maksymalnym Polish Princípio maximal de Hausdorff Portuguese
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, 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).