Exercise 3, § 2

a) Let K be a compact group and $K_{(n)}$ the set of elements of order $n$ of K. If $K\not= E_n$ for every $n$, then the set of elements of infinite order of K is dense in K. (If $K_{(n)}$ is of nonempty interior, one has $K_{(m)}= K$ for an integer $m\geqslant n.$)

b) Let Γ be a discrete commutative group whose elements are not of bounded order. Then Γ admits a countable quotient with the same property. (Reason as in part a) of exercise 2.)

c) If every neighbourhood of $e$ in G contains an element of infinite order, then G possesses a metrizable closed subgroup with the same property. (Reduce to the case where G is compact, and use a) and b).) Otherwise, and if G is not discrete, there exists an integer $q >1$ such that G contains a closed subgroup isomorphic to $(\mathbf{Z}/q\mathbf{Z})^{\mathbf{N}}$. (Use A, VII, p. 54, exercise 4, c).)