Particular Point Space is Irreducible

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be a particular point space.

Then $T$ is hyperconnected.

Proof
Trivially, by definition, every open set in $T$ contains $p$.

So:
 * $\forall U_1, U_2 \in \vartheta_p: p \in U_1 \cap U_2$

for $U_1, U_2 \ne \varnothing$.

Alternatively, from Closure of Open Set of Particular Point Space we have that:
 * $\forall U \in \vartheta_p: U \ne \varnothing \implies U^- = S$

where $U^-$ is the closure of $U$.

The result then follows from Hyperconnected iff Closure is Entire Space.