Let $G$ be a first countable Hausdorff topological group and $g\in G\setminus \{e\}$. Denote $[g]$ be the conjugacy class of $g$.
My question is that "Can $e\in \overline{[g]}$?", where $\overline{[g]}$ means the closure of $[g]$ in $G$.
Equivalently, "Does there exist a sequence $\{x_n\}$ in $G$ such that $x_n^{-1}gx_n\to e$ as $n\to \infty$?"
Or, we can just consider more explicit case that $G$ assume to be second countable locally compact Hausdorff group.
We already know that the answer is NO when the group $G$ is abelian or compact or discrete. But we can't find a solution for general cases.
Thanks in advance.