The Hausdorff maximal principle, (also called the Hausdorff maximality theorem) formulated and proved by Felix Hausdorff in 1914, is an alternate and earlier formulation of Zorn's lemma and therefore also equivalent to the axiom of choice. Felix Hausdorff Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ... 1914 (MCMXIV) was a common year starting on Thursday. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every non-empty partially ordered set in which every chain (i. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset (i.e. in a totally ordered subset which, if enlarged in any way, does not remain totally ordered: in general, there are many maximal totally ordered subsets containing a given totally ordered subset). In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...

An equivalent (but not obviously so) form of the theorem is that in every partially ordered set there exists a maximal totally ordered subset.