In a previous question I asked about the following version of the second isomorphism theorem for topological groups -I implicitly assume that every subgroup inherits the subspace topology-:
Let $G$ be a topological group and $H$ and $N$ subgroups. If $H$ is contained in the normalizer of $N$, there is a canonical continuous isomorphism $$\phi:\frac{H}{H\cap N}\rightarrow \frac{HN}{N}$$
In general we cannot guarantee that this $\phi$ is an homeomorphism. However, this is due to the fact that the continuos homomorphism $$H\rightarrow \frac{HN}{N}\text{,}$$ given by $h\mapsto hN$, cannot be guarantee to be open, close or and identification. And thus the first isomorphism theorem for topological groups does not give a homeomoprhism. Obviously when $H$ is open everything works nicely and we obtain a homeomorphism, are there more general (sufficient or necessary) conditions (on $H$ and $N$?) to guarantee that the previous map is open, closed or an identification?