# Is FC-center of a topologically finitely generated profinite group closed?

Let $$G$$ be a group. A element $$g\in G$$ is an FC-element if it has only finitely many conjugates in $$G$$. The set $$\Delta(G)$$ of FC-elements of $$G$$ is a characteristic subgroup of $$G$$, and it is called the FC-center of $$G$$.

Question: Suppose that $$G$$ is a topologically finitely generated profinite group. Is $$\Delta(G)$$ closed in $$G$$?

Note that it's not true without the condition "topologically finitely generated", see Page 1281 in Profinite Groups with Restricted Centralizers. Since $$G$$ is topologically finitely generated, the topology on $$G$$ should be determined by the algebraic structure, cf. On finitely generated profinite groups, I: strong completeness and uniform bounds. Hence, it seems that the answer should be Yes in our case. Any references would be appreciated.

All I know is the following: Firstly, note that $$\Delta(G)$$ is just the union of all centralisers $$C_G(H)$$ of all (abstract) subgroup $$H$$ of finite index. Here each $$H$$ must be open since $$G$$ is topologically finitely generated. Also, one can show that each $$C_G(H)$$ is closed. For each natural number $$n$$, it's well known that the number of open subgroups of $$G$$ of index $$n$$ is finite. What's the next step?

No. Consider the group $$\prod_{n\ge 5}\mathrm{Alt}(n)$$. Then it is topologically 2-generated. Its FC-center is $$\bigoplus_{n\ge 5}\mathrm{Alt}(n)$$, which is dense but not closed.