# center of a Hausdorff topological group is closed

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. $\square$

Title center of a Hausdorff topological group is closed CenterOfAHausdorffTopologicalGroupIsClosed 2013-03-22 18:01:48 2013-03-22 18:01:48 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 22A05