Editing Tychonoff's theorem (section)

== Proofs of Tychonoff's theorem ==

1) Tychonoff's 1930 proof used the concept of a [[complete accumulation point]].

2) The theorem is a quick corollary of the [[Alexander subbase theorem]].

More modern proofs have been motivated by the following considerations: the approach to compactness via convergence of subsequences leads to a simple and transparent proof in the case of countable index sets. However, the approach to convergence in a topological space using sequences is sufficient when the space satisfies the first axiom of countability (as metrizable spaces do), but generally not otherwise. However, the product of uncountably many metrizable spaces, each with at least two points, fails to be first countable. So it is natural to hope that a suitable notion of convergence in arbitrary spaces will lead to a compactness criterion generalizing sequential compactness in metrizable spaces that will be as easily applied to deduce the compactness of products. This has turned out to be the case.

3) The theory of convergence via filters, due to [[Henri Cartan]] and developed by [[Nicolas Bourbaki|Bourbaki]] in 1937, leads to the following criterion: assuming the [[ultrafilter lemma]], a space is compact if and only if each [[Ultrafilter (set theory)|ultrafilter]] on the space converges. With this in hand, the proof becomes easy: the (filter generated by the) image of an ultrafilter on the product space under any projection map is an ultrafilter on the factor space, which therefore converges, to at least one ''x<sub>i</sub>''. One then shows that the original ultrafilter converges to ''x''&nbsp;=&nbsp;(''x<sub>i</sub>''). In his textbook, [[James Munkres|Munkres]] gives a reworking of the Cartan–Bourbaki proof that does not explicitly use any filter-theoretic language or preliminaries.

4) Similarly, the [[Moore-Smith sequence|Moore–Smith]] theory of convergence via nets, as supplemented by Kelley's notion of a [[Net (mathematics)|universal net]], leads to the criterion that a space is compact if and only if each universal net on the space converges. This criterion leads to a proof (Kelley, 1950) of Tychonoff's theorem, which is, word for word, identical to the Cartan/Bourbaki proof using filters, save for the repeated substitution of "universal net" for "ultrafilter base".

5) A proof using nets but not universal nets was given in 1992 by Paul Chernoff.