Let $X = \prod_{j \in J} X_j$ be a D326: Cartesian product.
A D11: Set $\prod_{j \in J} E_j \subseteq X$ is a cylinder set in $X$ if and only if \begin{equation} \exists \, n \in \mathbb{N} : \exists \, j_0, \dots, j_n \in J : \forall \, j \in J \setminus \{ j_0, \dots, j_n \} : E_j = X_j \end{equation}

Let $X = \prod_{j \in J} X_j$ be a D326: Cartesian product.
Let $\mathcal{P}_{\mathsf{cofinite}}(J)$ be the D2200: Set of cofinite sets in $J$.
A D11: Set $\prod_{j \in J} E_j \subseteq X$ is a cylinder set in $X$ if and only if \begin{equation} \exists \, I \in \mathcal{P}_{\mathsf{cofinite}}(J) : \forall \, i \in I : E_i = X_i \end{equation}