Cantor Space is not Scattered

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Theorem

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.


Then $T$ is not scattered.


Proof

By definition, $T$ is scattered if and only if it contains no non-empty subset which is dense-in-itself.

We have that Cantor Space is Dense-in-itself.

Hence the result by definition of a scattered space.

$\blacksquare$


Sources