Cantor Space is Dense-in-itself

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 dense-in-itself.

Proof
Let $U \in \tau_d$ be open in $\mathcal C$.

Let $p \in U$.

Then $\exists x \in U: \exists \epsilon \in \R: d \left({x, p}\right) < \epsilon$.

Hence the result.