Cantor Space is Perfect

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 a perfect set.

Proof
From Cantor Space is Dense-in-itself, $\mathcal C$ contains no isolated points.

We also have that the Cantor Set is Closed in Real Number Space.

The result follows from the definition of perfect set.