Exercise 7, § 2

a) Show that if $G$ is a connected quasi-topological group (Exercise 5) then the commutator subgroup of $G$ is connected. (Let $P_k$ be the set of all products of $k$ commutators $x^{-1}y^{-1}xy$; show that $P_k$ is a union of connected sets each of which meets $P_{k-1}$).

b) Let $G$ be a non-commutative infinite group whose commutator subgroup is finite (for example, the product of a non-commutative finite group and an infinite commutative group). If $G$ carries the topology defined in Exercise 2 b) of § 1, show that $G$ is a connected semi-topological group whose commutator subgroup is not connected.