Totally Disconnected Space is Punctiform
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Theorem
Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected.
Then $T$ is punctiform.
Proof
Let $T = \struct {S, \tau}$ be totally disconnected.
Then by definition its components are singletons.
Thus by definition each of its connected sets are degenerate.
$\blacksquare$
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: Biconnectedness and Continua