Cantor Space is not Scattered

Theorem
Let $T = \left({\mathcal C, \tau_d}\right)$ be the Cantor space.

Then $T$ is not scattered.

Proof
By definition, $T$ is scattered 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.