Editing Tychonoff's theorem (section)

== Topological definitions ==

The theorem depends crucially upon the precise definitions of [[compact space|compactness]] and of the [[product topology]]; in fact, Tychonoff's 1935 paper defines the product topology for the first time. Conversely, part  of its importance is to give confidence that these particular definitions are the most useful (i.e. most well-behaved) ones.

Indeed, the Heine–Borel definition of compactness—that every covering of a space by open sets admits a finite subcovering—is relatively recent. More popular in the 19th and early 20th centuries was the [[Bolzano-Weierstrass]] criterion that every bounded infinite sequence admits a convergent subsequence, now called [[sequentially compact|sequential compactness]]. These conditions are equivalent for [[metrizable space]]s, but neither one implies the other in the class of all topological spaces.

It is almost trivial to prove that the product of two sequentially compact spaces is sequentially compact—one passes to a subsequence for the first component and then a subsubsequence for the second component. An only slightly more elaborate "diagonalization" argument establishes the sequential compactness of a countable product of sequentially compact spaces. However, the product of [[Continuum (set theory)|continuum]] many copies of the closed unit interval (with its usual topology) fails to be sequentially compact with respect to the product topology, even though it is compact by Tychonoff's theorem (e.g., see {{harvnb|Wilansky|1970|page=134}}).

This is a critical failure: if ''X'' is a [[completely regular]] [[Hausdorff space]], there is a natural embedding from ''X'' into [0,1]<sup>''C''(''X'',[0,1])</sup>, where ''C''(''X'',[0,1]) is the set of continuous maps from ''X'' to [0,1]. The compactness of [0,1]<sup>''C''(''X'',[0,1])</sup> thus shows that every completely regular Hausdorff space embeds in a compact Hausdorff space (or, can be "compactified".) This construction is the [[Stone–Čech compactification]]. Conversely, all subspaces of compact Hausdorff spaces are completely regular Hausdorff, so this characterizes the completely regular Hausdorff spaces as those that can be compactified. Such spaces are now called [[Tychonoff space]]s.