Product topology

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 natural-seeming, 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.

"Product space" redirects here. For other uses, see The Product Space.

Definition[edit]

Throughout, $I$ will be some non-empty index set and for every index $i\in I,$ let $X_{i}$ be a topological space. Denote the Cartesian product of the sets $X_{i}$ by

$X:=\prod X_{\bullet }:=\prod _{i\in I}X_{i}$

and for every index $i\in I,$ denote the $i$ -th canonical projection by

${\begin{aligned}p_{i}:\ \prod _{j\in I}X_{j}&\to X_{i},\\[3mu](x_{j})_{j\in I}&\mapsto x_{i}.\\\end{aligned}}$

The product topology, sometimes called the Tychonoff topology, on ${\textstyle \prod _{i\in I}X_{i}}$ is defined to be the coarsest topology (that is, the topology with the fewest open sets) for which all the projections ${\textstyle p_{i}:\prod X_{\bullet }\to X_{i}}$ are continuous. The Cartesian product ${\textstyle X:=\prod _{i\in I}X_{i}}$ endowed with the product topology is called the product space. The open sets in the product topology are arbitrary unions (finite or infinite) of sets of the form ${\textstyle \prod _{i\in I}U_{i},}$ where each $U_{i}$ is open in $X_{i}$ and $U_{i}\neq X_{i}$ for only finitely many $i.$ In particular, for a finite product (in particular, for the product of two topological spaces), the set of all Cartesian products between one basis element from each $X_{i}$ gives a basis for the product topology of ${\textstyle \prod _{i\in I}X_{i}.}$ That is, for a finite product, the set of all ${\textstyle \prod _{i\in I}U_{i},}$ where $U_{i}$ is an element of the (chosen) basis of $X_{i},$ is a basis for the product topology of ${\textstyle \prod _{i\in I}X_{i}.}$

The product topology on ${\textstyle \prod _{i\in I}X_{i}}$ is the topology generated by sets of the form $p_{i}^{-1}\left(U_{i}\right),$ where $i\in I$ and $U_{i}$ is an open subset of $X_{i}.$ In other words, the sets

$\left\{p_{i}^{-1}\left(U_{i}\right)\mathbin {\big \vert } i\in I{\text{ and }}U_{i}\subseteq X_{i}{\text{ is open in }}X_{i}\right\}$

form a subbase for the topology on $X.$ A subset of $X$ is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form $p_{i}^{-1}\left(U_{i}\right).$ The $p_{i}^{-1}\left(U_{i}\right)$ are sometimes called open cylinders, and their intersections are cylinder sets.

The product topology is also called the topology of pointwise convergence because a sequence (or more generally, a net) in ${\textstyle \prod _{i\in I}X_{i}}$ converges if and only if all its projections to the spaces $X_{i}$ converge. Explicitly, a sequence ${\textstyle s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }}$ (respectively, a net ${\textstyle s_{\bullet }=\left(s_{a}\right)_{a\in A}}$ ) converges to a given point ${\textstyle x\in \prod _{i\in I}X_{i}}$ if and only if $p_{i}\left(s_{\bullet }\right)\to p_{i}(x)$ in $X_{i}$ for every index $i\in I,$ where $p_{i}\left(s_{\bullet }\right):=p_{i}\circ s_{\bullet }$ denotes $\left(p_{i}\left(s_{n}\right)\right)_{n=1}^{\infty }$ (respectively, denotes $\left(p_{i}\left(s_{a}\right)\right)_{a\in A}$ ). In particular, if $X_{i}=\mathbb {R}$ is used for all $i$ then the Cartesian product is the space ${\textstyle \prod _{i\in I}\mathbb {R} =\mathbb {R} ^{I}}$ of all real-valued functions on $I,$ and convergence in the product topology is the same as pointwise convergence of functions.

Examples[edit]

If the real line $\mathbb {R}$ is endowed with its standard topology then the product topology on the product of $n$ copies of $\mathbb {R}$ is equal to the ordinary Euclidean topology on $\mathbb {R} ^{n}.$ (Because $n$ is finite, this is also equivalent to the box topology on $\mathbb {R} ^{n}.$ )

The Cantor set is homeomorphic to the product of countably many copies of the discrete space $\{0,1\}$ and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology.

Several additional examples are given in the article on the initial topology.

Every product of is T₀.

T₀ spaces

Every product of is T₁.

T₁ spaces

Every product of is Hausdorff.

Hausdorff spaces

Every product of is regular.

regular spaces

Every product of is Tychonoff.

Tychonoff spaces

A product of need not be normal.

normal spaces

Separation

Compactness

Connectedness

Metric spaces

Axiom of choice[edit]

One of many ways to express the axiom of choice is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.^[2] The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.

The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation,^[3] and shows why the product topology may be considered the more useful topology to put on a Cartesian product.

– space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology

Disjoint union (topology)

– Finest topology making some functions continuous

Final topology

– Coarsest topology making certain functions continuous - Sometimes called the projective limit topology

Initial topology

– Construction in category theory

Inverse limit

– A notion of convergence in mathematics

Pointwise convergence

– Topological space construction

Quotient space (topology)

– Inherited topology

Subspace (topology)

– Mathematical term

Weak topology

(1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.

Bourbaki, Nicolas

Willard, Stephen (1970). . Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796. Retrieved 13 February 2013.