Compact Complement Topology is not Ultraconnected

From ProofWiki
Jump to navigation Jump to search


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.


Proof 2

We have:

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

Hence the result.