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:

Compact Complement Topology is $T_1$
Non-Trivial Ultraconnected Space is not $T_1$

Hence the result.

$\blacksquare$


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