Linearly Ordered Space is T5/Mistake

Source Work

 * Part $\text {II}$: Counterexamples
 * Section $39$: Order Topology
 * Item $6$
 * Item $6$

Mistake

 * For each $\gamma$, select and fix some point $k_\gamma \in C_\gamma$. Then whenever $A_\alpha \cap \overline S_\alpha \ne \O$, there exists a unique ${k_\alpha}^+ \in {C_\alpha}^+$, the immediate successor of $A_\alpha$ ... otherwise, if $A_\alpha \cap {\overline S_\alpha}^\alpha = \O$, let $I_\alpha = \O$. ...

Correction
There is a minor typo here: there is a superfluous $\alpha$ superscript.

That last clause should say:
 * otherwise, if $A_\alpha \cap \overline S_\alpha = \O$, let $I_\alpha = \O$.

Also see

 * Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected