Totally Disconnected Space is T1

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space which is totally disconnected.

Then $T$ is a $T_1$ (Fréchet) space.

Proof
Let $T = \left({X, \vartheta}\right)$ be totally disconnected.

Then as its components are singletons, it follows that each of its points is closed.

From Equivalent Definitions for $T_1$ Space, it follows that $T$ is a $T_1$ (Fréchet) space.