Proposition 1

The closure $\overline{H}$ of a subgroup $H$ of a topological group $G$ is a subgroup of $G$. If $H$ is a normal subgroup of $G$, then so is $\overline{H}$.

If $a, b \in \overline{H}$ then $ab^{-1} \in \overline{H}$, because the mapping $(x, y) \to xy^{-1}$ is continuous on $G \times G$ and transforms $H \times H$ into $H$ (Chapter I, § 2, no. 1, Theorem 1). In the same way, the continuity of the mapping $x \to axa^{-1}$ shows that if $H$ is normal then $\overline{H}$ is normal.

In particular, the closure $N$ of the set $\{ e \}$ consisting only of the identity element of $G$, is a normal subgroup of $G$; and $N = \{ e \}$ if and only if $G$ is Hausdorff ($\S$ 1, no. 2, Proposition 2).