Countable Complement Space is Irreducible

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

Then $T$ is an irreducible space.

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

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

From Uncountable Subset of Countable Complement Space Intersects Open Sets, they intersect each other.

Hence the result from definition of irreducible space.