Exercise 10, § 1

Let $\mathbf{R}_d$ be the group $\mathbf{R}$ with the discrete manifold structure. Let H be the real Lie group $\mathbf{R} \times \mathbf{R}_d$. Let U be the set of elements of H of the form $(x, x)$ or of the form $(x, -x)$ where $x \in \mathbf{R}$. Then U is a Lie subgroup germ of H, U is discrete, the subgroup G of H generated by U is equal to H and the Lie group structure on G defined by the Corollary to Proposition 22 is the discrete group structure.