Hausdorff’s maximum principle

TheoremLet $X$ be a partially ordered set. Then there exists a maximal totallyordered subset of $X$ .

The Hausdorff’s maximum principle is one of the many theorems equivalentto theaxiom of choice (http://planetmath.org/AxiomOfChoice).The below proof uses Zorn’s lemma, whichis also equivalent to the.

Proof.

Let $S$ be the set of all totally ordered subsets of $X$ . $S$ is not empty, since the empty set is an element of $S$ . Partial order $S$ by inclusion. Let $\\tau$ be a chain (of elements) in $S$ . Being each totally ordered, the union of all these elements of $\\tau$ is again a totally ordered subset of $X$ , and hence an element of $S$ , as is easily verified. This shows that $S$ , ordered by inclusion, is inductive. The result now follows from Zorn’s lemma.∎

单词	HausdorffsMaximumPrinciple
释义	Hausdorff’s maximum principle TheoremLet $X$ be a partially ordered set. Then there exists a maximal totallyordered subset of $X$ . The Hausdorff’s maximum principle is one of the many theorems equivalentto theaxiom of choice (http://planetmath.org/AxiomOfChoice).The below proof uses Zorn’s lemma, whichis also equivalent to the. Proof. Let $S$ be the set of all totally ordered subsets of $X$ . $S$ is not empty, since the empty set is an element of $S$ . Partial order $S$ by inclusion. Let $\\tau$ be a chain (of elements) in $S$ . Being each totally ordered, the union of all these elements of $\\tau$ is again a totally ordered subset of $X$ , and hence an element of $S$ , as is easily verified. This shows that $S$ , ordered by inclusion, is inductive. The result now follows from Zorn’s lemma.∎
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