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.