Proposition 9.1.11.label Let $G$ be a topological group and $H \subset G$ be a subgroup, then $\ol H$ is also a subgroup.
Proof. By Proposition 5.5.3, for each $g \in G$, $g\ol H \subset \ol{gH}\subset \ol H$. Similarly, $\ol{H}^{-1}\subset \ol{H^{-1}}\subset \ol H$.$\square$
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