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.
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.
My test about how nice is a proof is: can I teach it to somebody just while standing in the queue at the canteen, on into subway car?
The one I like the most is the proof via ultrafilters. You only have to state the compactness of a topological space in terms of ultrafilters, which is easily obtained by the definition via open coverings.
X is compact if and only if every ultrafilter is convergent.
Then one observes that
1 any image of an ultrafilter is an ultrafilter (in particular, any projection from a product space)
2 any filter in the product space converges if and only if all its projections converge .
You really only need a few definitions and a few natural properties.
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.
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.
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.