Linearly Ordered Space is T5/Mistake

Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):

Part $\text {II}$: Counterexamples

Section $39$: Order Topology

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $39$. Order Topology: $6$