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I am reading 'Topology' by J.R. Munkres's first chapter on set theory. In the exercises 5-7 on page 72 he asks the reader to show that Zorn's lemma implies Hausdorff maximum principle via the following route : Zorn's lemma $\implies$ Kuratowski lemma $\implies$ Tukey lemma $\implies$ Maximum principle

What is wrong with the following proof of this statement ?

Proof : Let $X$ be a partially ordered set. Let $\mathcal{B}$ be the collection of strictly ordered subsets of $X$. Then $\mathcal{B}$ is partially ordered and satisfies the hypothesis of Zorn's lemma. Thus $\mathcal{B}$ has a maximal element. This element is a maximal strictly ordered subset of $X$ thereby completing the proof.

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The proof is fine. (Although you probably mean linearly ordered, not strictly ordered.)

I imagine that the route is suggested from a pedagogical point of view, so that you learn about a couple other equivalents to the axiom of choice and their typical uses.

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