Page "Hausdorff maximal principle" Paragraph 2

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.