Topological product

2020 Mathematics Subject Classification: Primary: 54-XX [MSN][ZBL]

The topological product (or Tikhonov product), of a family of topological spaces $\def\a{\alpha}\def\cT{\mathcal{T}} \{(X_\a,\cT_\a)\;|\; \a\in A\}$ is the topological space $(X,\cT)$ where $X$ is the Cartesian product (i.e. the complete direct product) of the sets $X_\a$ over $\a\in A$ and $\cT$ is the weakest (i.e. the smallest) topology (cf. Topological structure (topology)) on $X$ such that all the natural projection mappings $\pi_\a : (X,\cT) \to (X_\a,\cT_\a)$ are continuous. The topology $\cT$, moreover, is called the product topology, while the Tikhonov product $(X,\cT)$ is also said to be the topological product of the family of spaces $\{(X_\a,\cT_\a)\;|\; \a\in A\}$.

The standard base of the topological space $(X,\cT)$ is the family of all sets of the form $\pi_{a_1}^{-1}(U_{\a_1})\cap\cdots\cap \pi_{a_n}^{-1}(U_{\a_n})$, where $\a_1,\dots,\a_n$ is an arbitrary finite collection of elements of the index set $A$ and $U_{\a_i}$ is an arbitrary element of the topology $\cT_{\a_i}$, $i=1,\dots,n$.

In particular, if the family $\{(X_\a,\cT_\a)\;|\; \a\in A\}$ consists of two spaces $X$ and $Y$, then a base of the topology of the product $Z=X\times Y$ is formed by sets of the form $U\times V$, where $U$ is an arbitrary open set in $X$ and $V$ is an arbitrary open set in $Y$. A base of the topology of the product of an arbitrary finite ordered set of topological spaces can be described similarly. Examples of topological products are: the plane, which is the product of two straight lines; the $n$-dimensional space $\R^n$, which is the product of $n$ straight lines; and a torus, which is the product of two circles.

The earliest attempts to define the topological product of an infinite set of topological spaces was for the case of metrizable factors. Accordingly, attempts were made to describe the product topology in terms of convergence of ordinary (countable) sequences. When the family of factors is uncountable, it proved impossible to obtain the above result in this way, because the closure operation in the topological product of an uncountable set of non-singleton metrizable spaces cannot be described in the language of convergence of sequences, nor can it be reduced to the closure of countable sets.

The definition of the topological product of an infinite set of topological spaces was given by A.N. Tikhonov (1930). He also proved that the topological product of compact Hausdorff spaces is always a compact Hausdorff space (Tikhonov's theorem).

The construction of a topological product is one of the main tools in the formation of new topological objects from ones already existing. Using topological products one can construct a number of fundamental standard objects of general topology, in particular, a Tikhonov cube $I^\tau$, defined as the topological product of a family of cardinality $\tau$ of intervals of the real line. By Tikhonov's theorem, all Tikhonov cubes are compact. Tikhonov proved that every completely-regular $T_1$-space is homeomorphic to a subspace of a cube $I^\tau$.

In addition to the cubes $I^\tau$, an important role is played in topology by the spaces $D^\tau$ and $F^\tau$, which are the products of $\tau$ copies of spaces consisting, respectively, of two isolated points (the simple digon $D$) and of digons with one isolated point (the connected digon $F$). Every compact metric space is a continuous image of a Cantor perfect set, i.e. of a space homeomorphic to the product of a countable number of simple digons $D$; every zero-dimensional space, i.e. every $T_0$-space in which the open-closed sets form a base, is homeomorphic to a subset of a Cantor discontinuum $D^\tau$; every $T_0$-space is homeomorphic to a subset of the space $F^\tau$.

Related to the power and role of the operation of taking topological products, problems concerned with the behaviour of topological properties under formation of topological products are of central importance in general topology. The classes of Hausdorff spaces, regular spaces and completely-regular spaces are stable under the operation of taking topological products. However, the product of a normal space with an interval need not be a normal space. Even for a finite number of factors, important topological properties such as the Lindelöf property and paracompactness are not stable under taking topological products.

The theorem that follows is of importance in general topology and its applications (particularly in the construction of models of set theory): The Suslin number of the topological product of an arbitrary set of separable topological spaces is countable. In particular, the Suslin number of an arbitrary Tikhonov cube is countable.

By distinguishing certain subspaces in a full topological product, the operation of taking topological products gives rise to the useful operations of the $\Sigma$-product and the $\sigma$-product. The operation of taking the box product of topologies $\cT_\a$ leads to a completely-different topology on the Cartesian product of the sets $X_\a$.

Comments

A base for the box product consists of sets of the form $\prod_{\a\in A}U_\a$ with $U_\a$ open in $X_\a$ for every $\a$. For the ordinary product topology one demands that $U_\a=X_\a$ for all but finitely many $\a$. The box product is much more difficult to handle than the ordinary product.

The main current problems (in 1984) and then-recent results concerning topological products and $\Sigma$-products (respectively, box products) are surveyed in [KuVa], Chapt. 18 (respectively, Chapt. 4). See also [Do] for the box product.

The connected digon $F$ is in the West usually called the Sierpiński space. It is often denoted by $\bf 2$; the idea is that it is the ordered set $\{0,1\}$ with the $T_0$-topology naturally induced by the order, closed sets being lower sets.

References

[Al]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie", Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[ArPo]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises", Reidel (1984) (Translated from Russian) Zbl 0568.54001
[Bo]	N. Bourbaki, "Elements of mathematics. General topology", Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210 Zbl 0301.54002 Zbl 0301.54001 Zbl 0145.19302
[Do]	E.K. van Douwen, "Covering and separation properties of box products" G.M. Reed (ed.), Surveys in general topology, Acad. Press (1980) pp. 55–129
[En]	R. Engelking, "General topology", Heldermann (1989) MR1039321 Zbl 0684.54001
[KuVa]	K. Kunen (ed.) J.E. Vaughan (ed.), Handbook of set-theoretic topology, North-Holland (1984) MR0776619 Zbl 0546.00022