Compact Complement Space is not T2, T3, T4 or T5

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Theorem

Let $T = \left({\R, \tau}\right)$ be the compact complement topology on $\R$.


Then $T$ is not a $T_2$ (Hausdorff) space, $T_{2 \frac 1 2}$ (completely Hausdorff) space, $T_3$ space, $T_4$ space or $T_5$ space.


Proof

We have that a Compact Complement Space is a $T_1$ space.

From $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ space.


We have:

Compact Complement Topology is Irreducible
Irreducible Hausdorff Space is Singleton


From $T_{2 \frac 1 2}$ (completely Hausdorff) space is $T_2$ (Hausdorff) space, $T$ is not a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


From Regular Space is $T_2$ Space, $T$ is not a regular space.

By definition, a regular space is a space that is both a $T_0$ space and a $T_3$ space.

But $T$ is a $T_0$ space and not a regular space.

So it follows that $T$ can not be a $T_3$ space.


Next we have that a Normal Space is a $T_3$ Space.

But as $T$ is not a $T_3$ space, $T$ can not be a normal space.

By definition, a normal space is a space that is both a $T_1$ space and a $T_4$ space.

But $T$ is a $T_1$ space and not a normal space.

So it follows that $T$ can not be a $T_4$ space.


Finally we have that a $T_5$ Space is a $T_4$ Space.

But as $T$ is not a $T_4$ space, $T$ can not be a $T_5$ space.

$\blacksquare$


Sources