Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^\circ$ , $(\bar A)^\circ$ , $\overline{A^\circ}$ are pairwise distinct?
I found an example from a book that such a set exists. For example, consider $$A=[0,1)\cup (1,2]\cup(\mathbb Q\cap [3,4])\cup\{5\...
I have to prove that for any set $A\subseteq\mathbb{R}^n$,
$$ \large\overline{A^{\circ}} = \overline{\overline{A^{\circ}}^{\,\circ}} $$
This is what I got so far: for any set $A$ I'm using these definitions:
Interior:
$$\exists r > 0\text{ such that }\{x \mid B_r(x) \subseteq A\}$$
Closure:
$$...
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If $D$ is a closed set, what is the relation in general between the set $D$ and the closure of $\operatorname{Int}D$?
We know that $\operatorname{Int}D\subseteq D$, so $\overline{\operatorname{Int}D}\subseteq \overline{D}$, but since $D$ is closed, we have $\overline{D}=D$, so that $\overlin...
