Let $G$ be a group and $T$ the set of elements of finite order in $G$.
If $T$ is a subgroup of $G$, then $G/T$ is a torsion-free group.
Suppose $G$ is a compact Hausdorff topological group.
Is it true that $G/cl(T)$ is torsion-free? (where $cl(T)$ is the topological closure of $T$ in $G$).