Point in Particular Point Space is not Omega-Accumulation Point

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Theorem

Let $T = \left({S, \tau_p}\right)$ 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 $U = \left\{{x, p}\right\} \subseteq S$.

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 $H$.

$\blacksquare$


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