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.

$\blacksquare$

Also see

• Sierpiński Space is Ultraconnected

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $10$