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^{}.
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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

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$g{s}_{\lambda}\u27f6gs$

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${s}_{\lambda}g\u27f6sg$
But since $Z$ is the center of $G$ we have that $g{s}_{\lambda}={s}_{\lambda}g$, and as $G$ is Hausdorff one must have $sg=gs$. This implies that $s\in Z$, i.e. $Z$ is closed. $\mathrm{\square}$
Title  center of a Hausdorff topological group is closed 

Canonical name  CenterOfAHausdorffTopologicalGroupIsClosed 
Date of creation  20130322 18:01:48 
Last modified on  20130322 18:01:48 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  4 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 22A05 