Tychonoff's theorem

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In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paper of Eduard Čech.

Several texts identify Tychonoff's theorem as the single most important result in general topology [e.g. Willard, p. 120]; others allow it to share this honor with Urysohn's lemma.



The theorem crucially depends upon the precise definitions of 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 correct (i.e., most useful) 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 sequence admits a convergent subsequence, now called sequential compactness. These conditions are equivalent for metrizable spaces, but neither implies the other on 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 many copies of the closed unit interval fails to be sequentially compact.

This is a critical failure: if X is a completely regular Hausdorff space, there is a natural embedding from X into [0,1]C(X,[0,1]), where C(X,[0,1]) is the set of continuous maps from X to [0,1]. The compactness of [0,1]C(X,[0,1]) 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 spaces.


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