Corollary 1

If G is Hausdorff, the following conditions are equivalent: (i) G is solvable (resp. nilpotent); (ii) $\overline{D^iG} = \{e\}$ (resp. $\overline{C^iG} = \{e\}$) for sufficiently large $i$. $D^iG \subset \overline{D^iG}, C^iG \subset \overline{C^iG}$ and hence (ii) $\Rightarrow$ (i). $\{e\} = \{\overline{e}\}$ and hence (i) $\Rightarrow$ (ii) by Proposition 1.