Let $X$ and $Y$ be topological spaces. The topology on their Cartesian product $X\times Y$ has basis the products $U\times V$ of open sets $U\subset X$ and $V\subset Y$.
Given a (possibly infinite) set ${X_i}{i\in I}$ of topological spaces, the product topology on $\prod{i\in I} X_i$ is generated by the basis consisting of all products $\prod_{i\in I} U_i$ for $U$ open in $X$ and $U_i = X_i$ for all but finitely many $i\in I$.