Indiscrete Space is Irreducible

Theorem
Let $T = \struct {S, \set {\O, S} }$ 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.