Exercise 8, § 4

Every totally ordered set X, endowed with one of the topologies $\mathscr{T}_+(X), \mathscr{T}_-(X)$ (I, p. 91, Exercise 5) is completely normal. (If A, B are two subsets of X such that $A \cap \overline{B} = B \cap \overline{A} = \varnothing$, define, for every point $x \in A$, a neighbourhood $V_x$ of x, and for every $y \in B$, a neighbourhood $W_y$ of y such that $V_x \cap W_y = \varnothing$ for $x \in A$ and $y \in B$).