Sierpiński Space is Ultraconnected

Theorem
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

Then $T$ is ultraconnected.

Proof
The only closed sets of $T$ are $\O, \set 1$ and $\set {0, 1}$.

$\set 1$ and $\set {0, 1}$ are not disjoint.

Hence the result by definition of ultraconnected.

Also see

 * Particular Point Space is not Ultraconnected