Theorem - Let $G$ be a Hausdorff topological group. Then the center of $G$ is a closed normal subgroup.

$\,$

Proof: Let $Z$ be the center of $G$. We know that $Z$ is a normal subgroup of $G$. Let us see that it is closed.

Let $s\in\overline{Z}$, the closure of $Z$. There exists a net $\{s_{{\lambda}}\}$ in $Z$ converging to $s$. Then, for every $g\in G$, we have that

• $gs_{{\lambda}}\longrightarrow gs$

• $s_{{\lambda}}g\longrightarrow sg$

But since $Z$ is the center of $G$ we have that $gs_{{\lambda}}=s_{{\lambda}}g$, and as $G$ is Hausdorff one must have $sg=gs$. This implies that $s\in Z$, i.e. $Z$ is closed. $\square$