吉洪诺夫定理1930

对有限个空间来说，这个定理没有特别之处；对无限个，无论是可数无穷还是不可数无穷，这个结论仍然成立，它依赖于乘积拓扑的定义，与选择公理（它又等价于佐恩引理）是等价的。

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed 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 1935 article by Tychonoff, "Über einen Funktionenraum".

Tychonoff's theorem is one of the most foundational results in general topology. The theorem is also valid for topological spaces based on fuzzy sets.

Statement

Here is the statement of the theorem.

Strength

The above theorem turns out to be equivalent to the axiom of choice. There is a special case of the theorem saying

A product of compact Hausdorff spaces Xᵢ is compact Hausdorff. If, in addition, all the Xᵢ are nonempty, then the product ∏ Xᵢ is nonempty.

This Hausdorff version of the theorem is actually strictly weaker than the axiom of choice and is equivalent to the Boolean prime ideal theorem. (Note for an arbitrary product, the axiom of choice ensures the product is nonempty when each factor is nonempty. But in the Hausdorff case, the above version can ensure the nonempty-ness without Choice. In fact, it implies a version of the axiom of choice; see § Application: the Hahn–Banach theorem.)

Tychonoff's theorem for finite products

For a finite product of compact spaces, the theorem is substantially easier. For example, it would follow from

if X is a compact space and Y is any topological space, then the projection p:X× Y→ Y is proper; i.e., the pre-image of a compact set is compact.

Alternatively, the tube lemma can be used to give a direct proof; see Tube lemma § Examples and properties.

Further remark

For a metric space, compactness is equivalent to sequential compactness. Using Cantor's diagonal argument, it is straightforward to show a countable product of sequentially compact spaces is sequentially compact space; thus, for metric spaces at least, this gives the theorem for a countable product.

Applications

Tychonoff's theorem has been used to prove many other mathematical theorems. These include theorems about compactness of certain spaces such as the Banach–Alaoglu theorem on the weak-* compactness of the unit ball of the dual space of a normed vector space, and the Arzelà–Ascoli theorem characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence. They also include statements less obviously related to compactness, such as the De Bruijn–Erdős theorem stating that every minimal k-chromatic graph is finite, and the Curtis–Hedlund–Lyndon theorem providing a topological characterization of cellular automata.

Proofs

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.

3) The theory of convergence via filters, due to Henri Cartan and developed by Bourbaki, leads to the following criterion: assuming the ultrafilter lemma, a space is compact if and only if each 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 xi. One then shows that the original ultrafilter converges to x = (xi). See Filters in topology § Tychonoff's theorem for the ingredients used here.

Application: the Hahn–Banach theorem

A standard proof of the Hahn-Banach theorem in functional analysis uses the axiom of choice in the form of Zorn's lemma. Tychonoff's theorem for compact Hausdorff spaces, which is strictly weaker than the axiom of choice and is equivalent to the Boolean prime ideal theorem, can be used to give an alternative proof.

Specifically, Tychonoff's theorem in the Hausdorff case is used to deduce the following variant of the axiom of choice. Given a family of nonempty compact Hausdorff spaces Kₓ, x∈ X parametrized by a set X, by a choice function for it, we shall mean a function c:X→ U:=∪ _(x∈ X)Kₓ

such that c(x)∈ Kₓ for each x. Also, for a relation R U; i.e., a subset of U× U, we say such c R -consistent on a set S⊂ X (c(x),c(y))∈ R for each x,y∈ S.