Exercise 1, § 2

(a) For $G$ to have a countable basis, it is necessary and sufficient that $\widehat{G}$ have a countable basis. (If $G$ has a countable basis, then $L^1(G)$ has a countable basis, hence $\mathsf{X}(L^1(G))$ has a countable basis.) b) For $G$ to be countable at infinity, it is necessary and sufficient that $\widehat{G}$ admit an open subgroup with a countable basis, or equivalently that $\widehat{G}$ be metrizable. (Use prop. 3 of II, p. 248.)

c) Suppose that $G$ is compact. For $G$ to be metrizable, it is necessary and sufficient that $\widehat{G}$ be countable, or equivalently that $G$ be isomorphic to a closed subgroup of $\mathbf{T}^{\mathbf{N}}$. (A countable commutative discrete group is a quotient of $\mathbf{Z}^{(\mathbf{N})}.$)