Let $T_j = (X_j, \mathcal{T}_j)$ be a D1106: Topological space for each $j \in J$.
Let $X = \prod_{j \in J} X_j$ and $\mathcal{T} = \prod_{j \in J} \mathcal{T}_j$ each be a D326: Cartesian product.
Let $\mathcal{P}_{\mathsf{cofinite}}(J)$ be the D2200: Set of cofinite sets in $J$.
The set of open cylinder sets in $X$ with respect to $T = \{ T_j \}_{j \in J}$ is the D11: Set \begin{equation} \textstyle \{ \prod_{j \in J} U_j \subseteq \mathcal{T} \mid \exists \, I \in \mathcal{P}_{\mathsf{cofinite}}(J) : \forall \, i \in I : U_i = X_i \} \end{equation}