Hausdorff maximal principle

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset (a totally ordered subset that, if enlarged in any way, does not remain totally ordered). In general, there may be many maximal totally ordered subsets containing a given totally ordered subset.

An equivalent form of the Hausdorff maximal principle is that in every partially ordered set there exists a maximal totally ordered subset. To prove that this statement follows from the original form, let A be a partially ordered set. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \varnothing is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing ${\displaystyle \varnothing }$, hence in particular A contains a maximal totally ordered subset. For the converse direction, let A be a partially ordered set and T a totally ordered subset of A. Then

${\displaystyle \{S\mid T\subseteq S\subseteq A{\mbox{ and S totally ordered}}\}}$

is partially ordered by set inclusion ${\displaystyle \subseteq }$, therefore it contains a maximal totally ordered subset P. Then the set ${\displaystyle P}$ satisfies the desired properties.

The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is very similar to this proof.

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 (x₀, y₀) and (x₁, y₁) are two points of the plane ℝ², define (x₀, y₀) < (x₁, y₁) if y₀ = y₁ and x₀ < x₁. This is a partial ordering of ℝ² under which two points are comparable only if they lie on the same horizontal line. The maximal totally ordered sets are horizontal lines in ℝ².

• John Kelley (1955), General topology, Von Nostrand.
• Gregory Moore (1982), Zermelo's axiom of choice, Springer.
• James Munkres (2000), Topology, Pearson.