Closure of Open Set of Particular Point Space

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

Let $U \in \vartheta_p$ be open in $T$ such that $U \ne \varnothing$.

Then $U^- = S$ where $U^-$ denotes the closure of $U$.

Proof
By definition, $\forall U \in \vartheta_p, u \ne \varnothing: p \in U$.

From Limit Points in Particular Point Space, every point in $S$ is a limit point of $p$.

So by definition of closure, every point in $S$ is in $U^-$.