Compact Complement Topology is not Ultraconnected/Proof 1

Theorem
Let $T = \left({\R, \tau}\right)$ be the compact complement topology on $\R$.

Then $T$ is not an ultraconnected space.

Proof
By definition, closed sets of $T$ are compact sets of $T$.

So, for example, $\left[{0 \,.\,.\, 1}\right]$ and $\left[{2 \,.\,.\, 3}\right]$ are disjoint compact sets and therefore closed sets of $T$.

Hence the result by definition of ultraconnected.