# 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.).