Exercise 2, § 2

a) Every infinite commutative group Γ admits an infinite countable quotient group. (Embed an infinite countable subgroup Δ of Γ in a countable divisible group $\Delta '$ applying A, II, p. 185, exerc. 14; then extend to Γ the identical morphism of Δ into $\Delta '.$)

b) Every infinite compact commutative group contains an infinite closed metrizable subgroup. (Use a).)