Is it true that the center $Z$ of a topological group $G$ is closed?(maybe we need the space to be Hausdorff or something like that...) I was thinking I can just show it is opened. So if I pick $x\in Z$ then I need to find an open $U \ni x$ such that $U\subset Z$. But I am not sure how to show it.
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What's the centre of $O(3)$? – Angina Seng Jul 10 '20 at 14:47
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One needs some assumptions, e.g Hausdorffness of $G$, for this to be true. Otherwise any nonabelian group with trivial topology is a counter-example. – Moishe Kohan Jun 29 '24 at 23:45
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The centre of $G$ is the intersection of the centralisers of its elements. The centralizers are closed, so the centre is too.
In more detail $$Z(G)=\bigcap_{g\in C}C_G(g)$$ where $$C_G(g)=\{h\in G:ghg^{-1}h^{-1}=e\}$$ is closed in $G$.
Angina Seng
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1+1. And to fill in a step for the OP: to see that $C_G(g)$ is closed, show that the map $$\mu_g:h\mapsto ghg^{-1}h^{-1}$$ is continuous and therefore $C_G(g)=\mu_g^{-1}({e})$ is the continuous preimage of a closed set. – Noah Schweber Jul 10 '20 at 14:55
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If I'm not wrong, you need $G$ to be a Hausdorff space in order to $\{e\}$ be closed.
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This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post; instead, provide answers that don't require clarification from the asker. - From Review – RDK Jun 29 '24 at 23:35