Is the product of closed subgroups in topological group closed?

No, take $G$ is the group of real numbers $\Bbb R$, $H$ is $\Bbb Z$ and $K$ is $\Bbb Zx$ where $x$ is irrational. The group $H.K$ is dense.

No, this is not true. In the additive group of real numbers, the subgroups generated by $1$ and $\pi$ are closed, but their product is dense and not all of $\mathbb R$, so it is not closed.