Hausdorff's maximum principleHaudorffsMaximumPrinciple2002-09-29 19:48:232008-03-25 14:15:35Theorem% this is the default PlanetMath preamble. as your knowledge
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\makeatother{\bf Theorem}
Let $X$ be a partially ordered set. Then there exists a maximal totally
ordered subset of $X$.
The Hausdorff's maximum principle is one of the many theorems equivalent
to the
\PMlinkname{axiom of choice}{AxiomOfChoice}.
The below proof uses Zorn's lemma, which
is also equivalent to the
\PMlinkescapetext{axiom of choice}.
\begin{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.
\end{proof}