Particular Point Space is not Ultraconnected

Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space with at least three points.

Then $T$ is not ultraconnected.

Proof
Let $x, y \in S: x \ne p, y \ne p, x \ne y$.

Consider $\set x$ and $\set y$.

Neither are open as neither contain $p$.

So from Subset of Particular Point Space is either Open or Closed they are both closed.

We have that $\set x \cap \set y = \O$.

The result follows by definition of ultraconnected.

Also see

 * Sierpiński Space is Ultraconnected