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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.

Proof

We will use Zorn's lemma. Let PP be a proset and let CβŠ†PC \subseteq P be a chain. Consider the collection π’ž\mathcal{C} of chains in PP that contain CC, ordered by inclusion. If {C Ξ±} α∈AβŠ†π’ž\{C_\alpha\}_{\alpha \in A} \subseteq \mathcal{C} is a family totally ordered by inclusion, then the union ⋃ Ξ±C Ξ±\bigcup_\alpha C_\alpha, with the order coming from PP, is also totally ordered: any two elements x∈C Ξ±,y∈C Ξ²x \in C_\alpha, y \in C_\beta are comparable in max(C Ξ±,C Ξ²)max(C_\alpha, C_\beta). The hypotheses for Zorn’s lemma therefore obtain on π’ž\mathcal{C}, and we conclude that π’ž\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 βˆ…\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 x≀yx \leq y, then C=Cβˆͺ{y}C = C \cup \{y\} by maximality of CC; therefore y∈Cy \in C and hence y≀xy \leq x since xx is an upper bound of CC.

Revised on June 5, 2009 02:03:02 by Toby Bartels (169.235.52.169)