Excluded Point Space is not Irreducible

Theorem
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space with at least three points.

Then $T^*_{\bar p}$ is not irreducible.

Proof
By definition, open sets of $S$ are precisely the open sets of $S \setminus \set p$ under the discrete topology.

Let $x, y \in S \setminus \set p: x \ne y$.

Then $\set x$ and $\set y$ are both open sets of $T$ such that $\set x \cap \set y = \O$.

Hence the result, by definition of irreducible.

Also see

 * Sierpiński Space is Irreducible