Scattered T1 Space is Totally Disconnected

Theorem
Let $T = \left({X, \vartheta}\right)$ be a scattered topological space which is also a $T_1$ (Fréchet) space.

Then $T$ is totally disconnected.

Proof
Let $T = \left({X, \vartheta}\right)$ be a scattered space which is also a $T_1$ (Fréchet) space.