Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 9 / 1 / a / Solution

False. In the additive topological group $\mathbb{R}$, the subgroup $\mathbb{Z}$ is closed: every noninteger has a sufficiently small interval disjoint from it. It is not open, because no neighbourhood of an integer consists only of integers. Thus a closed subgroup need not be an open subgroup.