Exercise 4, § 4

Every linearly ordered set $X$ endowed with either of the topologies $\mathscr{O}_+(X), \mathscr{O}_-(X)$ (Chapter I, $§ 2$, Exercise 5) is completely normal. (If $A$ and $B$ are two subsets of $X$ such that $A \cap \overline{B} = B \cap \overline{A} = \varnothing$, define a neighbourhood $V_x$ of $x$ for each $x \in A$, and a neighbourhood $W_y$ of $y$ for each $y \in B$, such that $V_x \cap W_y = \varnothing$ for all $x \in A$ and all $y \in B$.)