Particular Point Space is not Weakly Countably Compact/Mistake

Source Work

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

Part $\text {II}$: Counterexamples

Section $8 \text { - } 10$: Particular Point Topology

Item $12$

Mistake

[A particular point space] $X$ is not weakly countably compact since any set which does not contain $p$ has no limit points.

However, this is true only of an infinite particular point space.

By definition, a space $X$ is weakly countably compact if and only if every infinite subset of $X$ has a limit point in $X$.

But a finite particular point space has no infinite subset.

So a finite particular point space is weakly countably compact vacuously.

Also see

• Infinite Particular Point Space is not Weakly Countably Compact

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $12$