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.

Definition

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

and for every index ${\displaystyle i\in I,}$ denote the ${\displaystyle i}$-th canonical projection by 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 ${\displaystyle U_{i}}$ is open in ${\displaystyle X_{i}}$ and ${\displaystyle U_{i}\neq X_{i}}$ for only finitely many ${\displaystyle i.}$ 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 ${\displaystyle X_{i}}$ 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 ${\displaystyle U_{i}}$ is an element of the (chosen) basis of ${\displaystyle X_{i},}$ 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 ${\displaystyle p_{i}^{-1}\left(U_{i}\right),}$ where ${\displaystyle i\in I}$ and ${\displaystyle U_{i}}$ is an open subset of ${\displaystyle X_{i}.}$ In other words, the sets

form a subbase for the topology on ${\displaystyle X.}$ A subset of ${\displaystyle X}$ is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form ${\displaystyle p_{i}^{-1}\left(U_{i}\right).}$ The ${\displaystyle p_{i}^{-1}\left(U_{i}\right)}$ 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 ${\displaystyle X_{i}}$ 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 ${\displaystyle p_{i}\left(s_{\bullet }\right)\to p_{i}(x)}$ in ${\displaystyle X_{i}}$ for every index ${\displaystyle i\in I,}$ where ${\displaystyle p_{i}\left(s_{\bullet }\right):=p_{i}\circ s_{\bullet }}$ denotes ${\displaystyle \left(p_{i}\left(s_{n}\right)\right)_{n=1}^{\infty }}$ (respectively, denotes ${\displaystyle \left(p_{i}\left(s_{a}\right)\right)_{a\in A}}$). In particular, if ${\displaystyle X_{i}=\mathbb {R} }$ is used for all ${\displaystyle i}$ then the Cartesian product is the space ${\textstyle \prod _{i\in I}\mathbb {R} =\mathbb {R} ^{I}}$ of all real-valued functions on ${\displaystyle I,}$ and convergence in the product topology is the same as pointwise convergence of functions.

Examples

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

The Cantor set is homeomorphic to the product of countably many copies of the discrete space ${\displaystyle \{0,1\}}$ 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.

Properties

The set of Cartesian products between the open sets of the topologies of each ${\displaystyle X_{i}}$ forms a basis for what is called the box topology on ${\displaystyle X.}$ In general, the box topology is finer than the product topology, but for finite products they coincide.

The product space ${\displaystyle X,}$ together with the canonical projections, can be characterized by the following universal property: if ${\displaystyle Y}$ is a topological space, and for every ${\displaystyle i\in I,}$ ${\displaystyle f_{i}:Y\to X_{i}}$ is a continuous map, then there exists precisely one continuous map ${\displaystyle f:Y\to X}$ such that for each ${\displaystyle i\in I}$ the following diagram commutes.

This shows that the product space is a product in the category of topological spaces. It follows from the above universal property that a map ${\displaystyle f:Y\to X}$ is continuous if and only if ${\displaystyle f_{i}=p_{i}\circ f}$ is continuous for all ${\displaystyle i\in I.}$ In many cases it is easier to check that the component functions ${\displaystyle f_i}$Zdroj:https://en.wikipedia.org?pojem=Product_topology
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