# Finite Complement Space is T1

## Theorem

Let $T = \struct {S, \tau}$ be a finite complement topology on any set $S$ which contains at least two points.

Then $T$ is a $T_1$ (Fréchet) space.

## Proof

Let $x, y \in S$.

Then:

$S \setminus \set x$ is an open set of $T$ containing $y$ but not $x$.
$S \setminus \set y$ is an open set of $T$ containing $x$ but not $y$.

Hence the result from definition of $T_1$ space.

$\blacksquare$