Definition 6

Given a filter $\mathfrak{F}_i$ on each set $X_i$ of a family of sets $(X_i)_{i \in I}$, the product of the filters $\mathfrak{F}_i$ is the filter on $X = \prod_{i \in I} X_i$ which has as a base the set of subsets of $X$ of the form $\prod_{i \in I} M_i$, where $M_i \in \mathfrak{F}_i$ for each $i \in I$ and $M_i = X_i$ for all but a finite number of indices. The product filter is denoted by $\prod_{i \in I} \mathfrak{F}_i$.

The reader may easily verify that the product of the filters $\mathfrak{F}_i$ can also be defined as the coarsest filter $\mathfrak{G}$ on $X$ such that $\mathrm{pr}_i(\mathfrak{G}) = \mathfrak{F}_i$ for each $i \in I$.

The preceding remarks show that if $\mathfrak{B}_i$ is a base of $\mathfrak{F}_i$ for each $i \in I$, then $\mathfrak{B}$ is a base of the product filter $\prod_{i \in I} \mathfrak{F}_i$ (no. 3, Proposition 3).

On a product $X = \prod_{i \in I} X_i$ of topological spaces, the neighbourhood filter of any point $x = (x_i)$ is the product of the neighbourhood filters of the $x_i$ (§ 4, no. 1).

The construction of a product filter $\mathfrak{F} = \prod_{i \in I} \mathfrak{F}_i$ is simpler when the index set $I$ is finite: a base of $\mathfrak{F}$ is then formed by all products $\prod_{i \in I} M_i$, where $M_i \in \mathfrak{F}_i$ for each $i \in I$. If $I = \{ 1, 2, \ldots, n \}$ we write

instead of $\prod_{i \in I} \mathfrak{F}_i$.