Infinite Particular Point Space is not Weakly Countably Compact

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Theorem

Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space.


Then $T$ is not weakly countably compact.


Proof

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


Let $H \subseteq S$ be an infinite subset of $S$ where $p \notin H$.

$H$ is not open in $T$ by definition.

So from Subset of Particular Point Space is either Open or Closed, $H$ is closed in $T$.

Then we have that a Closed Set in Particular Point Space has no Limit Points.

The result follows from definition of weakly countably compact.

$\blacksquare$


If $T = \left({S, \tau_p}\right)$ is a finite particular point space, then Finite Space Satisfies All Compactness Properties applies.


Mistakes in Sources

See 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology: $\text{II}: \ 8 - 10: \ 12$ where it is stated that:

Particular Point Space is not Weakly Countably Compact


Sources