Interior of Closed Set of Particular Point Space

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

Let $V \subseteq S$ be closed in $T$ such that $V \ne S$.

Then:
 * $V^\circ = \varnothing$

where $V^\circ$ denotes the interior of $V$.

Proof
By definition:
 * $\forall U \in \tau_p, U \ne \varnothing: p \in U$

Thus if $V$ is closed in $T$:
 * $\exists U \subseteq T: V = \complement_S \left({U}\right)$

So $p \notin V$.

Hence no open set of $T$ can be a subset of $V$ unless it is $\varnothing$.

Hence the result, by definition of interior.