Particular Point Space is not Ultraconnected

Theorem
Let $T = \left({S, \tau_p}\right)$ 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 $\left\{{x}\right\}$ and $\left\{{y}\right\}$.

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 $\left\{{x}\right\} \cap \left\{{y}\right\} = \varnothing$.

The result follows by definition of ultraconnected.

Also see

 * Sierpiński Space is Ultraconnected