Totally Disconnected but Connected Set must be Singleton

Theorem

Let $T = \struct {S, \tau}$ 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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness