Infinite Subset of Finite Complement Space Intersects Open Sets

Theorem
Let $T = \left({S, \tau}\right)$ be a finite complement topology on an infinite set $S$.

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

Then the intersection of $H$ with any non-empty open subset of $T$ is infinite.

Proof
Let $U \in \tau$ be any non-empty open subset of $T$.

Then $\complement_S \left({U}\right)$ is finite.

Suppose $H \cap U = \varnothing$.

Then from Intersection of Complement with Subset is Empty it follows that $H \subseteq \complement_S \left({U}\right)$ and so $H$ is finite.

So if $H$ is infinite it is bound to have a non-empty intersection with every open set in $T$.