Let $T_j = (X_j, \mathcal{T}_j)$ be a D1106: Topological space for each $j \in J$.
Let $X = \prod_{j \in J} X_j$ be a D326: Cartesian product.
Let $\mathsf{Cyl}_T (X)$ be the D3799: Set of open cylinder sets in $X$ with respect to $T = \{ T_j \}_{j \in J}$.
The product topology on $X$ with respect to $T = \{ T_j \}_{j \in J}$ is the D11: Set \begin{equation} \{ \cup E \mid E \subseteq \mathsf{Cyl}_T (X) \} \end{equation}