could any one just give hint for this one?
$G$ be a topological group such that $\forall x\in G, x\mapsto xy$ Homeomorphism, $H$ is a open subgroup of $G$, we need to prove $H$ is also closed
In a topological group the group multiplication is by definition continuous (and thus translations are homeomorphisms). You're probably trying to say that if $G$ is a group with topology such that right translations are homeomorphisms, then any open subgroup is also closed.
To show that, notice that $H$ is closed iff its complement is open, which you can write out explicitly using the group operations.