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What's the "shortest path" to Tychonoff's theorem (the product of compact spaces is compact)?

Of course, I don't expect that anyone will spell this out. I'm just looking for a sketch of the main stages along the way. I can then "connect the dots".

By "shortest path" I mean the quickest, most direct path to a proof of Tychonoff's theorem for someone who knows the basics of set theory (unions, intersections, complements, Cartesian products, projections), knows what a topology is (open and closed sets, bases, and subbases), and has the requisite mathematical acumen.

I have looked at various textbooks on general topology for this, but in all of them Tychonoff's theorem is positioned as a pinnacle of sorts, and I even get the impression that the authors use the "long march" towards this theorem as an expository device to introduce a lot of machinery, much of which gets used, of course, in the theorem's eventual proof. I'm hoping that a more direct proof is possible if one doesn't have such an agenda.

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  • $\begingroup$ Shortest path is to assume the Tychonoff theorem, of course! :-) $\endgroup$ – Asaf Karagila Dec 30 '13 at 10:31
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    $\begingroup$ @kjo: A nice proof is given in Principles of Harmonic Analysis (Deitmar, Echterhoff), First Edition, Theorem A.7.2, p. 273. The proof is a bit different in the Second Edition. $\endgroup$ – Watson Aug 24 '16 at 9:19
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For someone who has not previously been exposed to filters, probably the shortest path is by way of the Alexander subbase theorem; the link gives both a fairly complete sketch of the proof of this theorem and the very easy proof from it of the Tikhonov product theorem.

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I very much like sequences as main tools in metric spaces. I thus like nets as main tools in topology. The American Mathematical Monthly published Paul R. Chernoff's "A Simple Proof of Tychonoff's Theorem Via Nets" (jstor). In less than two pages the basic ingredients are presented and the theorem proved.

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  • $\begingroup$ i haven't read that. Does it use Kelley's theorem on universal nets (which is itself a bit on the tricky side)? $\endgroup$ – dfeuer May 8 '13 at 23:00
  • $\begingroup$ no, Chernoff's proof does not use universal nets. $\endgroup$ – Ittay Weiss May 9 '13 at 0:19
  • $\begingroup$ But the basic ingredients are presented without proof... $\endgroup$ – Michael Greinecker Dec 30 '13 at 10:24
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I just scanned for you two sheets (1, 2) of a small book “Combinatorics of Numbers” by Ihor Protasov. Here you can go from the definition of a topological space to the proof of Tychonov Theorem. The proof is based on the notion of an ultrafilter.

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  • $\begingroup$ The links seem to be dead. (Perhaps they would last longer if they were embedded as images.) $\endgroup$ – Martin Sleziak Dec 29 '13 at 21:13
  • $\begingroup$ @MartinSleziak Thanks. I corrected the links. $\endgroup$ – Alex Ravsky Dec 30 '13 at 4:12
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My favourite one is the one which uses the fact that: A space is compact if every net has convergent subnet - If you feel comfortable with nets, try the proof suggested by Ittay Weiss.

Otherwise, the standard proof which uses Zorn's Lemma for a maximal cover without finite subcover is not that bad after all.

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