Sierpiński Space is Ultraconnected

Theorem
Let $T = \left({\left\{{0, 1}\right\}, \tau_0}\right)$ be a Sierpiński space.

Then $T$ is ultraconnected.

Proof
The only closed sets of $T$ are $\varnothing, \left\{{1}\right\}$ and $\left\{{0, 1}\right\}$.

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

Hence the result by definition of ultraconnected.

Also see

 * Particular Point Space is Not Ultraconnected