Openness preserved for product with normal subgroup

Question

For a group $P$ with open subgroup $Q$ and a group $N$ such that $P$ normalizes $N$ and $P\cap N$ is closed in $P$, I wonder whether $QN$ is an open subgroup of $PN$.

Answer 1

Recently, I come up with a solution as following: For quotient map $\phi: P\to P/P\cap N$, since $\phi$ is an open map, $\phi(Q)\cong QN/N$ is open in $P/P\cap N\cong PN/N$. On the other hand, considering quotient map $\psi:PN\to PN/N$, it is straightforwardly derived that $QN$ is open in $PN$.

I wonder whether that's correct.