Finite T1 Space is Discrete/Proof 2
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Theorem
Let $S$ be a finite set.
Let $T = \left({S, \tau}\right)$ be a $T_1$ (Fréchet) space.
Then $\tau$ is the discrete topology on $S$.
Proof
We have from Finite Complement Topology is Minimal $T_1$ Topology that $\tau$ space is an expansion of a finite complement space.
But from the definition, any such finite complement space on a finite set is discrete.
But as the Discrete Topology is Finest Topology, $\tau$ must itself be the discrete topology on $S$.
$\blacksquare$