> [!definition] > > Let $\bracs{X_i, \fF_i}_{i \in I}$ be a family of sets and [[Filter|filters]]. The **product filter** $\prod_{i \in I}\fF_i$ is the filter on $\prod_{i \in I}X_i$ generated by the cylinder sets > $ > \mathfrak C(\seqi{\fF}) = \bracs{\pi_i^{-1}(E_i): i \in I, E_i \in \fF_i} > $ > where > > 1. $\prod_{i \in I}\fF_i$ is the smallest filter on $\prod_{i \in I}X_i$ such that $\pi_i\paren{\prod_{i \in I}\fF_i} = \fF_i$ for all $i \in I$. > 2. Finite intersections of cylinder sets form a base for $\prod_{i \in I}\fF_i$. > 3. If each $\fB_i$ is a base for $\fF_i$, then $\mathfrak C(\seqi{\fB})$ is a base for $\prod_{i \in I}\fF_i$.