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 $T$.

By definition, $S$ is infinite.

By definition, $\relcomp S {U_1}$ and $\relcomp S {U_2}$ are finite.

From Complement of Finite Subset of Infinite Set is Infinite, it follows that $U_1$ and $U_2$ are both infinite.

From Infinite Subset of Finite Complement Space Intersects Open Sets:
 * $U_1 \cap U_2\ne \O$

That is, $U_1$ and $U_2$ intersect each other.

As $U_1$ and $U_2$ are arbitrary, the result follows by definition of irreducible space.