Totally Disconnected Space is Punctiform

Theorem

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

Then $T$ is punctiform.

Proof

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

Then by definition its components are singletons.

Thus by definition each of its connected subsets are degenerate.

$\blacksquare$

Sources

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