# Compact Complement Topology is not Ultraconnected

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## 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.

$\blacksquare$

## Proof 2

We have:

Hence the result.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 22: \ 3$