Cantor Space is Compact

Theorem
Let $\mathcal C$ be the Cantor set.

Let $\left({\R, \tau_d}\right)$ be the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\mathcal C$ is a compact subset of $\left({\R, \tau_d}\right)$.

Proof
We have Cantor Set Closed in Real Number Space.

Taking, for example, $0 \in \mathcal C$ and $1 \in \R$ it is clear that:
 * $\forall x \in \mathcal C: d \left({0, x}\right) \le 1$

and so $\mathcal C$ is bounded.

Hence the result from the Heine-Borel Theorem.