Center of a topological group is closed?

Question

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.

Answer 1

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 $G$.