# Countable Complement Space is not T3, T4 or T5

## Theorem

Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.

Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space.

## Proof

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

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

We then have that a Countable Complement Space is not $T_2$.

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 $T$ cannot 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, $T$ cannot 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 $T$ cannot 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, $T$ cannot 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: $20$. Countable Complement Topology: $1$