Corollary

Let $(x_i)_{i \in I}$ be a summable family in a commutative group whose identity element has a countable fundamental system of neighbourhoods. Then the set of indices $i$ such that $x_i \neq o$ is countable.

Let $(V_n)$ be a countable fundamental system of neighbourhoods of $o$. If $H_n$ is the set of all indices $i$ such that $x_i \notin V_n$, then the set $H$ of indices $i$ such that $x_i \neq o$ is the union of the sets $H_n$, and each of the $H_n$ is finite by Proposition 1.

This corollary is no longer necessarily valid if we do not assume that the origin has a countable fundamental system of neighbourhoods. \* Consider, for example, the product group $\mathbf{R}^\mathbf{R}$ [the additive group of all finite real-valued functions of a real variable, with the topology of pointwise convergence (cf. Chapter X, , no. 3)], and let $f_a$ be the element of $\mathbf{R}$ such that $f_a(a) = 1$ and $f_a(x) = 0$ if $x \neq a$; then the family $(f_a)_{a \in \mathbf{R}}$ is summable and its sum is the function whose value at each point of $\mathbf{R}$ is 1. \*