Closure in Infinite Particular Point Space is not Compact

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

Let $A \in \vartheta_p$ be open in $T$.

Let $A^-$ be the closure of $A$.

Then $A^-$ is not compact.

Proof
From Closure of Open Set of Particular Point Space, we have that $A^- = S$.

The result follows from Infinite Particular Point Space is Not Compact.