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 1937 paper of Eduard Čech.

- Topological definitions
- Applications
- Proofs of Tychonoff's theorem
- Tychonoff's theorem and the axiom of choice
- Proof of the axiom of choice from Tychonoff's theorem
- See also
- Notes
- References
- External links

Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma).^{ [1] } The theorem is also valid for topological spaces based on fuzzy sets.^{ [2] }

The theorem depends crucially 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 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 sequence admits a convergent subsequence, now called sequential compactness. These conditions are equivalent for metrizable spaces, 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 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 Wilansky 1970 , p. 134).

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.

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.

As a rule of thumb, any sort of construction that takes as input a fairly general object (often of an algebraic, or topological-algebraic nature) and outputs a compact space is likely to use Tychonoff: e.g., the Gelfand space of maximal ideals of a commutative C* algebra, the Stone space of maximal ideals of a Boolean algebra, and the Berkovich spectrum of a commutative Banach ring.

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 Bourbaki in 1937, 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 *x _{i}*. One then shows that the original ultrafilter converges to

4) Similarly, the Moore–Smith theory of convergence via nets, as supplemented by Kelley's notion of a 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.

All of the above proofs use the axiom of choice (AC) in some way. For instance, the third proof uses that every filter is contained in an ultrafilter (i.e., a maximal filter), and this is seen by invoking Zorn's lemma. Zorn's lemma is also used to prove Kelley's theorem, that every net has a universal subnet. In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice in **ZF**. Note that one formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, the proof cannot proceed along such straightforward lines. Thus Tychonoff's theorem joins several other basic theorems (e.g. that every vector space has a basis) in being *equivalent* to AC.

On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). A first glance at the second proof of Tychnoff may suggest that the proof uses no more than (BPI), in contradiction to the above. However, the spaces in which every convergent filter has a unique limit are precisely the Hausdorff spaces. In general we must select, for each element of the index set, an element of the nonempty set of limits of the projected ultrafilter base, and of course this uses AC. However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using (BPI), and in fact the converse also holds. Studying the *strength* of Tychonoff's theorem for various restricted classes of spaces is an active area in set-theoretic topology.

The analogue of Tychonoff's theorem in pointless topology does not require any form of the axiom of choice.

To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty. The trickiest part of the proof is introducing the right topology. The right topology, as it turns out, is the cofinite topology with a small twist. It turns out that every set given this topology automatically becomes a compact space. Once we have this fact, Tychonoff's theorem can be applied; we then use the finite intersection property (FIP) definition of compactness. The proof itself (due to J. L. Kelley) follows:

Let {*A _{i}*} be an indexed family of nonempty sets, for

Now define the cartesian product

along with the natural projection maps *π _{i}* which take a member of

We give each *X _{i}* the topology whose open sets are the cofinite subsets of

and prove that these inverse images are nonempty and have the FIP. Let *i _{1}*, ...,

By the FIP definition of compactness, the entire intersection over *I* must be nonempty, and the proof is complete.

- ↑ Stephen Willard "General Topology" Dover Books ISBN 978-0-486-43479-7 pp 120
- ↑ Joseph Goguen "The Fuzzy Tychonoff Theorem" Journal of Mathematical Analysis and Applications Volume 43, Issue 3, September 1973, pp 734-742

In mathematics, the **axiom of choice**, or **AC**, is an axiom of set theory equivalent to the statement that *a Cartesian product of a collection of non-empty sets is non-empty*. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related areas of mathematics, a **metrizable space** is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is . **Metrization theorems** are theorems that give sufficient conditions for a topological space to be metrizable.

In mathematics, more specifically in general topology and related branches, a **net** or **Moore–Smith sequence** is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space.

In topology and related areas of mathematics, a **product space** is the Cartesian product of a family of topological spaces equipped with a natural topology called the **product topology**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In the mathematical field of order theory, an **ultrafilter** on a given partially ordered set (poset) *P* is a certain subset of *P,* namely a maximal filter on *P*, that is, a proper filter on *P* that cannot be enlarged to a bigger proper filter on *P*.

In topology and related branches of mathematics, a **normal space** is a topological space *X* that satisfies **Axiom T _{4}**: every two disjoint closed sets of

In mathematics, a **paracompact space** is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

In the mathematical discipline of general topology, **Stone–Čech compactification** is a technique for constructing a universal map from a topological space *X* to a compact Hausdorff space *βX*. The Stone–Čech compactification *βX* of a topological space *X* is the largest, most general compact Hausdorff space "generated" by *X*, in the sense that any continuous map from *X* to a compact Hausdorff space factors through *βX*. If *X* is a Tychonoff space then the map from *X* to its image in *βX* is a homeomorphism, so *X* can be thought of as a (dense) subspace of *βX*; every other compact Hausdorff space that densely contains *X* is a quotient of *βX*. For general topological spaces *X*, the map from *X* to *βX* need not be injective.

The **Baire category theorem** (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, the **Boolean prime ideal theorem** states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals, or distributive lattices and *maximal* ideals. This article focuses on prime ideal theorems from order theory.

In general topology, a branch of mathematics, a non-empty family *A* of subsets of a set *X* is said to have the **finite intersection property** (FIP) if the intersection over any finite subcollection of *A* is non-empty. It has the **strong finite intersection property** (SFIP) if the intersection over any finite subcollection of *A* is infinite.

In topology, a **subbase** for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In the mathematical field of set theory, **Martin's axiom**, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, , behave roughly like . The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.

In the mathematical field of set theory, an **ultrafilter** is a *maximal proper filter*: it is a filter on a given non-empty set which is a certain type of non-empty family of subsets of that is not equal to the power set of and that is also "maximal" in that there does not exist any other proper filter on that contains it as a proper subset. Said differently, a proper filter is called an ultrafilter if there exists *exactly one* proper filter that contains it as a subset, that proper filter (necessarily) being itself.

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- Tychonoff's theorem at ProofWiki
- Mizar system proof: http://mizar.org/version/current/html/yellow17.html#T23

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