Compact Complement Topology is not Ultraconnected

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

Then $T$ is not an ultraconnected space.

Proof 1
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.

Proof 2

 * Compact Complement Topology is $T_1$


 * Non-Trivial Ultraconnected Space Cannot be $T_1$