Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 5 / 1 / i / Solution

If $H$ is an open subgroup, each left coset $gH$ is open because left translation is a homeomorphism in a topological group. Its complement is the union of all the other cosets,This is open, so $H$ is closed. No compactness or separation assumption is required.