Limit Points in Particular Point Space

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

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

Then $x$ is a limit point of $p$.

Similarly, let $U \subseteq S$ such that $p \in U$.

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

Then $x$ is a limit point of $U$.

Proof
Every open set of $T = \left({S, \vartheta_p}\right)$ except $\varnothing$ contains the point $p$ by definition.

So every open set $U \in \vartheta_p$ such that $x \in U$ contains $p$.

So:
 * by definition of the limit point of a set, $x$ is a limit point of $U$, and


 * by definition of the limit point of a point, $x$ is a limit point of $p$.