Totally Disconnected but Connected Set must be Singleton

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$ be both totally disconnected and connected.

Then $H$ is a singleton.

Proof
If $H$ is totally disconnected then its individual points are separated.

If $H$ is connected it can not be represented as the union of two (or more) separated sets.

So $H$ can have only one point in it.