Open subgroup

An open subgroup is also closed, since its complement is the union of its other open cosets. Conversely a closed subgroup need not be open, as $\mathbb{Z}\subset \mathbb{R}$ shows.

An open normal subgroup $N$ of a topological group is simultaneously open in its topology and invariant under conjugation. For a profinite group, the quotient $G/N$ is a finite discrete group: its cosets form a disjoint open cover, so compactness gives finitely many. Conversely the group homomorphism kernel of a continuous homomorphism to a finite discrete group is open and normal. Such subgroups form an identity-neighbourhood basis of every profinite group.