Is cantor set open when intersecting a closed interval $[0,\frac{1}{3}]$ (in cantor set)？

Question

In our class, our professor said cantor set is clopen (both closed and open). One argument is that the interior of a cantor set intersecting with a closed interval, say $A = \Delta \cap [0,1/3]$ whose interior is $A$ itself. From my point of view, the interior should be empty. Is there any difference if we considered $A$ as a subset of $\Delta$(cantor set itself) rather than $\mathbb{R}$?

Answer 1

Let $X$ be a metric space; let $A \subset X$; let $B \subset A$; then $B$ is called open (resp. closed) in $A$ if and only if there is some $C$ open (resp. closed) in $X$ such that $B = C \cap A$. From this definition it follows that the Cantor set is clopen in itself.

However, the Cantor set is not clopen in $\mathbb{R}$, as you have remarked.