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, I {\displaystyle I} will be some non-empty index set and for every index i ∈ I , {\displaystyle i\in I,} let X i {\displaystyle X_{i}} be a topological space. Denote the Cartesian product of the sets X i {\displaystyle X_{i}} by
X := ∏ X ∙ := ∏ i ∈ I X i {\displaystyle X:=\prod X_{\bullet }:=\prod _{i\in I}X_{i}}
and for every index i ∈ I {\displaystyle i\in I} , denote the i {\displaystyle i} -th canonical projection by
p i : ∏ j ∈ I X j → X i , ( x j ) j ∈ I ↦ x i . {\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}}}
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