Closure of Infinite Subset of Finite Complement Space

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

Let $H \subseteq S$ be an infinite subset of $S$.

Then $H^- = S$ where $H^-$ is the closure of $S$.

Proof
Let $H$ be an infinite subset of $S$.

From Limit Points of Infinite Subset of Finite Complement Space, every point of $S$ is a limit point of $H$.

Hence the result from the definition of closure.