Cantor Space is not Scattered

Theorem
Let $\left({\mathcal C, \tau_d}\right)$ be the Cantor set considered as a topological subspace of the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\mathcal C$ is not scattered.

Proof
From Cantor Space is Dense-in-itself, $\mathcal C$ it itself dense-in-itself.

Hence the result by definition of a scattered space.