Linearly Ordered Space is T5/Mistake
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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 $6$
- Section $39$: Order Topology
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
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $39$. Order Topology: $6$