Exercise 6, § 2

Let $G$ be the orthogonal group $O_2(\mathbf{Q})$, identified with a subspace of the space $\mathbf{Q}^4$ of all $2 \times 2$ matrices over $\mathbf{Q}$. Show that the topology induced on $G$ by the topology of $\mathbf{Q}^4$ is compatible with the group structure of $G$. Show that the commutator subgroup of $G$ is not closed and that its closure in $G$ is the group $SO_2(\mathbf{Q})$. \*