Totally Disconnected but Connected Set must be Singleton
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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