Let $(G,\mathcal T)$ be a metrizable topological group and $C$ be a dense countable subgroup of $G$. How large can be the cardinality of $G$? Can it have a larger cardinality than $|\Bbb R|$?
It seems that Jones' Lemma cannot be used because $C$ need not be discrete.
If $X$ is any first-countable Hausdorff space and $C\subseteq X$ is a countable dense subset, then every element of $X$ is a limit of some sequence in $C$. Since there are only $|C|^{|\mathbb{N}|}\leq|\mathbb{R}|$ different sequences in $C$ and each sequence has at most one limit, $X$ can have at most $|\mathbb{R}|$ different points.