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. $\\square$

单词	CenterOfAHausdorffTopologicalGroupIsClosed
释义	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. $\\square$
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