Exercise 2, § 3

Let $G$ be a linearly ordered group (not necessarily commutative), with more than one element. Then the topology $\mathscr{T}_0(G)$ (Chapter I, , Exercise 5) is compatible with the group structure of $G$. If $G$ is connected in this topology, show that $G$ is isomorphic to the additive group $\mathbf{R}$ (use Exercise 7 of Chapter IV, § 2, and Proposition 2 of § 2)