Exercise 5, § 4

Every linearly ordered set $X$, endowed with the topology $\mathscr{O}_u(X)$ (Chapter I, $§ 2$, Exercise 5), is completely normal. [Consider first the case where $X$ is compact; show with the help of Chapter IV, $§ 2$, Exercise 6 that every open set in $X$ is the union of mutually disjoint open intervals; use this result to prove the proposition, by considering (in the notation of Exercise 3) the complement of the closed set $\overline{A} \cap \overline{B}$, and then the complement of $\overline{B}$; to pass to the general case, in which $X$ is arbitrary, use Chapter IV, $§ 4$, Exercise 7.]