= Open normal subgroup
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>.
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