Is there a name or perhaps some interesting equivalent condition for the following condition on an abelian topological group $G$ with uniformity generated by the neighbourhoods of 0?
Every point in the completion of $G$ is in the closure of a countable subset of $G$ (note: not necessarily the same countable subset for every point)
(Not assuming $G$ is metrizable)
If $G$ is first-countable, then the definition of $\hat{G}$ can be made by Cauchy sequences, and the result is immediate. In the case when $G$ is not first countable, the definition of the completion is more difficult to work with and I don't see the proof immediately, but I would be surprised if the property you are interested in held in the non-first-countable case.