TITLE: Open subgroups of a topological group are closed
QUESTION [18 upvotes]: Let $G$ be a topological group such that for each $x \in G$ the mapping $x\mapsto xy$ is a  homeomorphism. If $H$ is a open subgroup of $G$, prove that $H$ is also closed

Could anyone just give hint for this one?

REPLY [10 votes]: 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.