Exercise 9, § 1

Let I be the set of $(x, \sqrt{2}x) \in \mathbf{R}^2$ for $-\frac{1}{2} < x < \frac{1}{2}$. Let U be the canonical image of I in the real Lie group $H = \mathbf{T}^2$. Show that U is a Lie subgroup germ of H but that the subgroup $H'$ of H generated by U is dense in H and not closed in H.