(This question has been on math.SE for over a week and has not gotten any answers.)

Let $G$ be a (T$_0$) topological abelian group, and let $0$ be its identity element.

Assume that for all index sets $I$ and all functions $f\colon I\to G$, if

$\big[$for each neighborhood $U$ of $0$, $f(i)\in U$ for all but finitely many $i$ $\big]$

then $\: $$\displaystyle\sum_{i\in I}\hspace{.03 in}f(i)$$ \:$ exists.

Does it follow that every neighborhood of $0$ contains an open subgroup $H$ of $G$?