Compact Complement Topology is not Ultraconnected/Proof 2

Theorem

Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.

Then $T$ is not an ultraconnected space.

Proof

We have:

Compact Complement Topology is $T_1$

Non-Trivial Ultraconnected Space is not $T_1$

Hence the result.

$\blacksquare$