Compact Complement Space is not T2, T3, T4 or T5
Theorem
Let $T = \struct {\R, \tau}$ 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:
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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $22$. Compact Complement Topology: $2$