Compact Complement Topology is not Ultraconnected

Theorem

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

$\blacksquare$

Proof 2

We have:

Compact Complement Topology is $T_1$

Non-Trivial Ultraconnected Space is not $T_1$

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $22$. Compact Complement Topology: $3$