Exercise 3, § 7

a) Show that every totally disconnected compact group is the inverse limit of a family of finite discrete groups (use Exercise 18 of § 4).

b) Let $G$ be a totally disconnected compact group and let $L$ be a closed subgroup of $G$. Show that there is a continuous section $G/L \to G$ associated with the canonical mapping $G \to G/L$. [Use a) and Proposition 3 of § 3; for each $\alpha$ consider the finite set $F_\alpha$ of sections $G_\alpha / L_\alpha \to G_\alpha$, and remark that these sets from an inverse system with respect to the canonical surjective mappings $h_{\alpha\beta} : F_\beta \to F_\alpha$].