Exercise 3, § 3

Let $G$ be a totally ordered group (not necessarily commutative); if, when $G$ is provided with the topology $\mathscr{T}_0(G)$, $G$ is locally compact and non-discrete, then $G$ is locally isomorphic to $\mathbf{R}$, and the neutral component of $G$ is an open subgroup isomorphic to $\mathbf{R}$ (use Exercise 6 of IV, p. 47, and Proposition 2 of V, p. 9).