Totally Disconnected Space is T1

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Theorem

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

Then $T$ is a $T_1$ (Fréchet) space.


Proof

Let $T = \left({S, \tau}\right)$ be totally disconnected.

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

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

$\blacksquare$


Sources