Particular Point Space is Irreducible/Proof 1

Proof
By definition, $T = \left({S, \tau}\right)$ is irreducible every two non-empty open sets of $T$ have non-empty intersection.

Let $U_1$ and $U_2$ be non-empty open sets of $T$.

By definition of particular point space, $p \in U_1$ and $p \in U_2$.

Thus:
 * $p \in U_1 \cap U_2$

and so:
 * $U_1 \cap U_2 \ne \varnothing$

Hence the result.