What does the quotient topological group $\mathbb{R}/(\mathbb{Z} + \sqrt{2}\mathbb{Z})$ look like? It is not Hausdorff; in fact it is first-countable and every sequence converges to every element. Is there a name for this property?
More generally, what can we say about $G/H$, where $G$ is a topological group and $H$ a dense normal subgroup thereof?
$G/{H}$ where $H$ is dense normal is an indiscrete (or trivial) space; the only open subsets are the whole space and $\emptyset$. This is indeed first countable and every sequence converges to every element, even the constant ones.
This is a very uninteresting space and that's why we normally take the quotient wrt closed normal subgroups, because that ensures us that the quotient is $T_1$ and thus Hausdorff and completely regular etc. So we then have more theory to work with.