Indiscrete Space is Irreducible

Theorem
Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Then $T$ is irreducible.

Proof
There is only one non-empty open set in $T$.

So there can be no two open sets in $T$ which are disjoint.

Hence (trivially) $T$ is irreducible.