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This question was formerly posted on MSE https://math.stackexchange.com/questions/4578923/ without getting an answer.

I know that Tychonov's theorem, Zorn's lemma, the axiom of choice, the well-ordering theorem and many other results are logically equivalent in the sense that either of them implies all other in ZF. But this is not my question.

I would like to know if there is a direct proof of Zorn's lemma using Tychonov's theorem for a suitably constructed produt of compact spaces.

I am sure that everybody understands the question although it is hard to formalize what I want. But again, this is not the point of this question.

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    $\begingroup$ In general waiting a single day before cross posting is frowned upon. People are busy, questions take time to answer. $\endgroup$
    – Asaf Karagila ♦
    Commented Nov 19, 2022 at 11:20
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    $\begingroup$ I strongly agree with Asaf. In particular since your question got several upvotes and constructive comments as well, including an idea by Arturo (which you didn't answer). Please be more patient, it will probably be answered there. The statement "... without getting an answer" sounds inappropriate to me after waiting just one day. $\endgroup$
    – Martin Brandenburg
    Commented Nov 19, 2022 at 11:29
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    $\begingroup$ I apologize. My experience on MSE is that questions are quickly drowning if they are not answered within a day. Moreover, since even experts like you don't know the answer immediately, the question seems to be more research level than I previously thought. $\endgroup$
    – Jochen Wengenroth
    Commented Nov 19, 2022 at 11:33
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    $\begingroup$ A more serious obstruction seems to be that most direct proofs of Zorn's lemma are fairly complicated and are, arguably, a repackaging of the simple proof based on the well-ordering theorem. $\endgroup$
    – Michael Greinecker
    Commented Nov 19, 2022 at 11:47
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    $\begingroup$ Kelly's proof of Tychonov $\Rightarrow$ AC is simple. This suggests that Tychonov $\Rightarrow$ Zorn should not be much easier to prove than AC $\Rightarrow$ Zorn. If there is a straightforward proof of Tychonov $\Rightarrow$ Zorn, I would change the way I do the various equivalents to AC when I teach real analysis. $\endgroup$
    – Bill Johnson
    Commented Nov 19, 2022 at 17:07

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