Exercise 9, § 4

Every totally ordered set X, endowed with the topology $\mathscr{T}_0(X)$ (I, p. 91, Exer. 5) is completely normal. (Consider first the case where X is compact; first prove that every open set in X is the union of a set of pairwise disjoint open intervals, using IV, p. 47, Exer. 2; use this result to prove the proposition, by considering (with the notations of Exer. 7) the complement of the closed set $\overline{A} \cap \overline{B}$, then the complement of the set $\overline{B}$. To pass to the general case where X is arbitrary, use IV, p. 52, Exer. 7).