Topological properties on Cantor set

Question

Show that the cantor set with the usual topology on $R$ is a normal , regular subspace. It seems to me that is true as the cantor set is subset in $[0,1] \subset{R}$ which is a metric space . So it has to satisfy the properties as R does. I will be glad if you suggest me proofs for that (and if there is a source in this topic).

Answer 1

All metric spaces are (perfectly) normal and the Cantor set $C$ is a metric space, as subspace of a metric space ($[0,1]$ or $\Bbb R$, whatever). It's a simple as that, indeed. Or use that a compact Hausdorff space is normal (and thus regular). Depending on what you've covered so far.