Exercise 6, § 2

If G is a torsion group, then G and $\widehat{G}$ are totally disconnected. ($\widehat{G}$ is totally disconnected according to Cor. 2 of II, p. 250; for proving that G is totally disconnected, reduce to the case where G is compact using Prop. 3 of II, p. 248, and then prove that $\widehat{G}$ is a torsion group.)