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
Follows directly from:


 * Particular Point Topology is Closed Extension Topology of Discrete Topology


 * Closure of Open Set of Closed Extension Space