Dispersion Point in Particular Point Space

Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.

Then $p$ is dispersion point of $T$.

Proof
Let $H = S \setminus \set p$.

Let $T_H = \struct {H, \tau_H}$ be the topological subspace induced on $H$ by $\tau_p$.

From Particular Point Space less Particular Point is Discrete, the space $T_H$ is discrete.

We have Discrete Space is Locally Connected.

Thus from Totally Disconnected and Locally Connected Space is Discrete we have that $S \setminus \set p$ is totally disconnected.

Hence the result, from definition of dispersion point.