Editing Hausdorff maximal principle (section)

== Examples ==
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.

If (x<sub>0</sub>, y<sub>0</sub>) and (x<sub>1</sub>, y<sub>1</sub>) are two points of the plane <math>\mathbb{R}^{2}</math>, define (x<sub>0</sub>, y<sub>0</sub>) < (x<sub>1</sub>, y<sub>1</sub>) if y<sub>0</sub> = y<sub>1</sub> and x<sub>0</sub> < x<sub>1</sub>. This is a partial ordering of <math>\mathbb{R}^{2}</math> under which two points are comparable only if they lie on the same horizontal line. The maximal totally ordered sets are horizontal lines in <math>\mathbb{R}^{2}</math>.