Finite Complement Space is Irreducible

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

Then $T$ is irreducible.

Proof
Let $U_1, U_2 \in \tau$ be non-empty open sets of $T$.

We have that $U_1$ and $U_2$ are both infinite if $S$ is.

From Infinite Subset of Finite Complement Space Intersects Open Sets, they intersect each other.

Hence the result from definition of irreducible space.