# Totally Disconnected Space is Punctiform

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## Theorem

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

Then $T$ is punctiform.

## Proof

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

Then by definition its components are singletons.

Thus by definition each of its connected sets are degenerate.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$: Biconnectedness and Continua