Let $C$ be the usual Cantor defined $C = \bigcap F_n$ where $F_1 = [0,1]$, $F_2 = [0,\frac{1}{3}]\cup[\frac{2}{3},1]$, $\dots$. I am trying to prove some elementary properties about this set and was wondering if anyone had any input about my proofs.
- $C$ is closed.
- Every point in $C$ is a boundary point of $C$.
- $C^0 = \emptyset$, where $C^0$ denotes the interior of $C$.
(1): I think this one is straightforward. $C$ is an intersection of closed sets, and the arbitrary intersection of closed sets is always closed, so $C$ is closed.
(2): I think I understand the spirit of this proof, but I feel like my argument is not good towards the end. Let $x \in C$, then clearly for any $r > 0$ we have $B_x(r)\cap C \neq \emptyset$, since $x$ belongs to both. To show that the intersection with the complement is not empty, notice that each $F_n$ is made up of a union of $2^n$ closed intervals all of the length $\frac{1}{3^{n-1}}$. Therefore we can find large enough $N$ so that $\frac{1}{3^{N-1}} < r$. Hence $B_x(r)$ contains points outside of $C$. Therefore $x$ is a boundary point of $C$.
(3): This proof I feel like its more or less the same as the second part of (2).
Any advice is appreciated!