Countable Complement Space is not T2

Theorem

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

Then $T$ is not a $T_2$ (Hausdorff) space.

Proof

We have:

Countable Complement Space is Irreducible

Irreducible Hausdorff Space is Singleton

$\blacksquare$

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: $1$