Limit Points in Closed Extension Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_p = \left({S^*_p, \tau^*_p}\right)$ be the closed extension space of $T$.

Let $x \in S$.

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

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

Let $x \in S$.

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

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

So every open set $U \in \tau^*_p$ 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 $p$.