Dispersion Point of Excluded Point Space

Theorem
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.

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

Proof
We have that the Excluded Point Topology is Open Extension Topology of Discrete Topology.

So $S \setminus \set p$ is a discrete space.

Then a discrete space is totally disconnected.

The result follows from definition of dispersion point.