Point in Particular Point Space is not Omega-Accumulation Point

Theorem

Let $T = \struct {S, \tau_p}$ be a particular point space.

Let $x \in S$ such that $x \ne p$.

Let $H \subseteq S$ such that $p \in H$.

Then $x$ is not an $\omega$-accumulation point of $H$.

Proof

Let $x \in S, x \ne p$.

By Limit Points in Particular Point Space, $x$ is a ‎limit point of $H$.

Consider the set $U = \set {x, p} \subseteq S$.

By definition of the particular point topology, $U$ is open in $T$.

But as $U$ contains only $x$ and $p$, it is clear that $U$ does not contain an infinite number of points of $H$.

Hence, by definition, $x$ can not be an $\omega$-accumulation point of $H$.

$\blacksquare$

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: $3$