If we want to iterate a commutative and associative law of composition over $\{a_i\}_{i \in I}$ in some topological space for an index set $I$, there is a simple way to do it using the directed set of finite subsets $F \subset I$. The composition over $I$ can then be written as the limit over the net $\prod_{u \in F} a_i$ if it exists:
$$\prod_{i\in I} a_i := \lim_{F\subset I} \prod_{i \in F} a_i$$
In the countable sum case (over $\mathbb C$) this is different from a normal series in that it converges iff the series converges absolutely.
The construction allows you to consider uncountable sums and products: An example of an uncountable sum would be in the strong operator topology of a non-seperable Hilbertspace, if we have mutually orthogonal projections $\pi_i$ indexed by $\mathbb R$ with joint kernel $\{0\}$, then $\sum_{i \in \mathbb R} \pi_i =\mathbb 1$ in this topology.
Is there a similarly convenient way to get limits of sums and products using filters instead of nets?