Definition 1

Let $(x_i)_{i \in I}$ be a family of points of a Hausdorff commutative group $G$; let $\mathfrak{F}(I)$ be the set of finite subsets of the index set $I$, and for each finite subset $J$ of $I$, let $s_J$ be the sum of those $x_i$ such that $i \in J$. The family $(x_i)_{i \in I}$ is said to be summable if the mapping $J \to s_J$ has a limit with respect to the section filter $\Phi$ of the set $\mathfrak{F}(I)$ directed by the relation $\subset$; this limit is then said to be the sum of the family $(x_i)_{i \in I}$ and is denoted by $\sum_{i \in I} x_i$ (or simply $\sum_i x_i$, or even $\sum x_i$, when there is no risk of ambiguity).

Definition 1 is equivalent to the following: the family $(x_i)$ is summable and its sum is $s$ if, for each neighbourhood $V$ of the origin in $G$, there is a finite subset $J_0$ of $I$ such that for each finite subset $J \supset J_0$ of $I$ we have $s_J \in s + V$.

If $G$ is written multiplicatively, and if we put $p_J = \prod_{i \in J} x_i$ for each finite subset $J$ of $I$, the family $(x_i)$ will be said to be multipliable if the mapping $J \to p_J$ has a limit with respect to the filter $\Phi$; this limit is called the product of the family $(x_i)$, and is denoted by $\prod_{i \in I} x_i$.