Finite Complement Space is not T2

Theorem

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

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

Proof

We have:

Finite Complement Space is Irreducible

Irreducible Hausdorff Space is Singleton

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: Exercise $4.3: 1$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $18 \text { - } 19$. Finite Complement Topology: $6$