# 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

$\blacksquare$