Hausdorff maximal principle

The Hausdorff maximal principle is a version of Zorn's lemma, equivalent to the usual version and thus (given excluded middle) equivalent to the axiom of choice.

Statement and proofs

Given a poset (or proset) SS, let a chain in SS be a subset AA of SS which, as a sub-proset, is totally ordered. A chain AA is maximal (as a chain) if the only chain that AA is contained in is AA itself.

Theorem (Hausdorff maximal principle)

Every chain in a proset is contained in a maximal chain.


We will use Zorn's lemma. Let PP be a proset and let CPC \subseteq P be a chain. Consider the collection 𝒞\mathcal{C} of chains in PP that contain CC, ordered by inclusion. If {C α} αA𝒞\{C_\alpha\}_{\alpha \in A} \subseteq \mathcal{C} is a family totally ordered by inclusion, then the union αC α\bigcup_\alpha C_\alpha, with the order coming from PP, is also totally ordered: any two elements xC α,yC βx \in C_\alpha, y \in C_\beta are comparable in max(C α,C β)max(C_\alpha, C_\beta). The hypotheses for Zorn’s lemma therefore obtain on 𝒞\mathcal{C}, and we conclude that 𝒞\mathcal{C} has a maximal element, which is clearly maximal in the collection of all chains.

Proof of converse

Conversely, suppose that the Hausdorff maximal principle holds; we will prove Zorn’s lemma. Suppose given a poset (or preorder) PP such that every chain in PP has an upper bound. Since \empty is a chain, the Hausdorff maximal principle implies that PP contains a maximal chain CC; let xx be an upper bound of CC. Then xx is maximal: if xyx \leq y, then C=C{y}C = C \cup \{y\} by maximality of CC; therefore yCy \in C and hence yxy \leq x since xx is an upper bound of CC.

Last revised on June 5, 2009 at 02:03:02. See the history of this page for a list of all contributions to it.