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.

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.

$\blacksquare$

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $13 \text { - } 15$. Excluded Point Topology: $3$