Find and instable subset of $\mathbb{R}$ with respect to closure and interior Let be $(\mathbb{R},\tau_E)$, with $\tau_E$ the euclidean topology. Find a set $A\subseteq \mathbb{R}$ such that $A,\overline{A},(\overline{A})^o,\overline{(\overline{A})^o},A^o,\overline{A^o},(\overline{A^o})^o$ are all different.
I'm struggling with this because intervals of the form $[a,b)$ aren't working, because, for example, the interior of the closure of $[a,b)$ is equal to the interior of $[a,b)$. I thought about something like $[a,b)\cap \mathbb{Q}$, but it also doesn't work. I didn't try with others unions or intersections because closure and interior don't behave very well with that (for example $\overline{A}\cap\overline{B} \neq \overline{A\cap B}$), and so they aren't easy to determine.
Any suggestions?

$A = \{1,\frac12, \frac13, \frac14,\ldots\} \cup (2,3) \cup (3,4) \cup \{4\frac12\} \cup [5,6] \cup ([7,8)\cap \Bbb Q)$ will do, e.g. So a combination of several ideas (intervals with gap, rational intervals, singletons, etc.).
(credits: Steen and Seebach, Counterexamples in Topology p.61 )
