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

• $g{s}_{\lambda }⟶gs$

• $s_{\lambda }g⟶sg$

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	2013-03-22 18:01:48
Last modified on	2013-03-22 18:01:48
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	4
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 22A05