Open Subgroup is Closed

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Theorem

Let $G$ be a topological group.

Let $H\leq G$ be an open subgroup.

Then $H$ is closed.

Proof

By Right and Left Regular Representations in Topological Group are Homeomorphisms, the left cosets of $H$ are open.

By Left Coset Space forms Partition, the complement of $H$ is a union of left cosets of $H$.

Because the complement of $H$ is open, $H$ is closed.

$\blacksquare$