Cantor Space is Dense-in-itself

Theorem
Let $T = \struct {\mathcal C, \tau_d}$ be the Cantor space.

Then $T$ is dense-in-itself.

Proof
Let $U \in \tau_d$ be open in $T$.

Let $p \in U$.

Then $\exists x \in U: \exists \epsilon \in \R: \map d {x, p} < \epsilon$.

Hence the result.