Give an example of subset $B$ of the real line $\mathbb{R}$ so the subsets $A$, $Int(A)$, $\overline{A}$, dont intersect

Question

What is an example of a subset $A$ of the real line $\mathbb{R}$ (equipped with the standard metric topology), such that the subsets $A$, $Int(A)$, $\overline{A}$, $\overline{Int(A)}$ and Int($\overline{A}$) are pairwise different?

Answer 1

Let $B=\Bbb Q\cap(1,2),$ and let $A=(0,1)\cup B.$

Answer 2

Take $A=((0,1) \cap \mathbb{Q}) \cup (2,3)$. The interior of A is $(2,3)$, the closure of A is $[0,1] \cup [2,3]$, the closure of the interior of A is $[2,3]$, and the interior of the closure of A is $(0,1) \cup (2,3)$.