A counter-example of the second isomorphism theorem for topological groups

Question

Let $G$ be a topological group and $H$ and $N$ subgroups. Suppose $H$ is contained in the normalizer of $N$, then by using arguments of the second isomorphism theorem we can show that there is a canonical continuous isomorphism $$\phi:H/H\cap N\rightarrow HN/N$$ Are there cases in which this fails to be an actual homeomorphism?

Answer 1

Presumably $H$ and $HN$ take the subspace topology from $G$ and project it onto the quotient groups?

Would the following be a counterexample? Let $G=\mathbb{R}$ be the additive group of reals, $H=\mathbb{Z}\cdot\sqrt2$ and $N=\mathbb{Q}$. Then $H\cap N$ is trivial, so $H/(H\cap N)$ inherits the discrete topology from $H$. On the other hand $N$ is dense in $G$, so $HN/N$ has only trivial closed sets, i.e. it has the trivial topology.