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 $S$.

$\blacksquare$