I am learning some stuff about the interior, closure and boundary of sets $A\subset\mathbb R^n$ and I am wondering about the following:
1) $\partial\partial A=\partial A$ ?
2) $\partial\partial\partial A=\partial A$ ?
3) $\partial\partial A=\emptyset$ ?
So 1) is false for e.g. $A=\mathbb Q$ with $\partial A=\mathbb R\neq\emptyset=\partial\partial A$
2) and 3) seems kinda hard. I guess 3) is wrong but I don't have a counterexample.
So does anybody have an idea about 2) and 3) ?
Add: A point $x$ is a boundary point of a set $A\subset \mathbb{R}^n$ if every neighborhood of $x$ contains a point of $A$ and $A^c$.
Best Answer
For the second question, when $A = \mathbb{Q}$, $\partial A = \mathbb{R}$ and $\partial\partial A = \emptyset $ and $\partial\partial\partial A = \emptyset$ as well.
For the third, example, consider the open interval $A = (-1,1)$ then $\partial A = \{-1,1\}$ and $\partial\partial A = \{-1,1\} \neq \emptyset$.