Particular Point Space is Scattered

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be a particular point space.

Then $T$ is scattered.

Proof
Let $H \subseteq S$ such that $p \notin H$.

From Subsets of Particular Point Space Either Open or Closed, $H$ is closed in $T$.

We have that Closed Set in Particular Point Space has No Limit Points.

So if $p \notin H$ then $H$ has no limit points.

So from Point is Isolated Iff Not a Limit Point every element of such a set $H$ is an isolated point.

So if $p \notin H$ then $H$ is by definition not dense-in-itself.

Now let $H \subseteq S$ such that $p \in H$.

We have that $\left\{{p}\right\} \in \vartheta_p$.

That is, $p$ is an open point of $T$.

From Open Points are Isolated, $p$ is an isolated point.

So if $p \in H$ then $H$ is by definition not dense-in-itself.

Hence the result, by definition of a scattered space.