Where is first-published proof of Tychonoff's product theorem using ultrafilters? Henri Cartan's paper "Filtres et ultrafFiltres et ultrafiltres" in Comptes Rendus, vol. 205 (1937), gives the result that a space is (quasi-)compact if and only if each ultrafilter on the space converges there.
That is one of the key assertions needed to prove the Tychonoff product theorem. (The two others are: (i) the image of an ultrafilter under a surjection is itself an ultrafilter; and (ii) an ultrafilter on a product space converges if and only if its image under each projection onto a factor space converges.)
Yet Cartan does not seem to apply his result in the cited paper to the Tychonoff theorem.
I know that Bourbaki's "Topologie générale" does give the ultrafilter proof of Tychonoff's theorem.
Is that the first place where this proof is published? And if not, where might it be?
 A: I think your best bet is to look it up in the Handbook of the History of General Topology.
Tychonoff proved it first for spaces of the form $[0,1]^m$ using complete accumulation points (1930, and in 1935 a more general version), using a transfinite induction argument IIRC. Engelking uses families of closed sets with the FIP, assumes it has empty intersection, then uses Teichmüller-Tukey to extend it to a maximal such family of closed sets and derives a contradiction from that. According to his notes the proof follows a 1941 paper by Chevalley and Frink. Tukey at that time had already given a proof (1940, in Convergence and uniformity in topology) using "ultraphalanxes" (seems close to ultranets from what I could find online), but they wanted to avoid that notion. They also note that their method extends to products of $H$-closed spaces as well.
Probably the ultrafilter idea is inspired on this proof. Willard uses ultranets instead, and Munkres' proof is similar to Engelking's, using a maximal family of sets with the FIP. So probably Bourbaki (i.e. one of its members) was the first to use filters and ultrafilters in an explicit way, but I'm not sure when such a proof was first published.
