\begin{definition}[Definition:Generalized Sum]
Let $\struct {G, +}$ be a commutative topological semigroup.
Let $\family {g_i}_{i \mathop \in I}$ be an indexed subset of $G$.
Consider the set $\FF$ of finite subsets of $I$.
Let $\subseteq$ denote the subset relation on $\FF$.
By virtue of Finite Subsets form Directed Set, $\struct {\FF, \subseteq}$ is a directed set.
Define the net:
:$\phi: \FF \to G$
by:
:$\ds \map \phi F = \sum_{i \mathop \in F} g_i$
Then $\phi$ is denoted:
:$\ds \sum \set {g_i: i \in I}$
and referred to as a '''generalized sum'''.
Statements about '''convergence''' of $\ds \sum \set {g_i: i \in I}$ are as for general convergent nets.
\end{definition}