Excluded Point Space is not Irreducible

Theorem
Let $T = \left({S, \tau_{\bar p}}\right)$ 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 $S \setminus \left\{{p}\right\}$ under the discrete topology.

Let $x, y \in S \setminus \left\{{p}\right\}: x \ne y$.

Then $\left\{{x}\right\}$ and $\left\{{y}\right\}$ are both open sets of $T$ such that $\left\{{x}\right\} \cap \left\{{y}\right\} = \varnothing$.

Hence the result, by definition of irreducible.

Also see

 * Sierpiński Space is Irreducible