Topological closure of subgroup is closed subgroup

Statement
Suppose $$G$$ is a topological group and $$H$$ is a subgroup of $$G$$. Denote by $$\overline{H}$$ the closure of $$H$$ in $$G$$ as a subset of a topological space. Then, $$\overline{H}$$ is a closed subgroup of $$G$$, i.e., it is a subgroup as well as a closed subset of $$G$$.

Related facts

 * Topological closure of subgroup belonging to a subvariety of the variety of groups is in the same subvariety