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