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}