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.

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

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

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

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

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. It is the initial topology on ${\textstyle \prod _{i\in I}X_{i}}$ with respect to the family of projections ${\displaystyle \left\{p_{i}\mathbin {\big \vert } i\in I\right\}}$. 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

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