Modified Fort Space is not T2

Theorem
Let $T = \left({S, \tau_{a, b}}\right)$ be a modified Fort space.

Then $T$ is not a $T_2$ (Hausdorff) space.

Proof
Consider $U, V \in \tau_{a, b}$ such that $a \in U, b \in V$.

We have that both $U$ and $V$ are cofinite.

So $U$ and $V$ must be infinite.

Suppose $U \cap V = \varnothing$.

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

But this contradicts the fact that $U$ is infinite.

So $U \cap V \ne \varnothing$.

Hence the result by definition of $T_2$ (Hausdorff) space.