Compact Complement Topology is not Ultraconnected/Proof 1

Theorem

Let $T = \struct {\R, \tau}$ 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.

$\blacksquare$