Playing with closure and interior to get the maximum number of sets Can you find $A \subset \mathbb R^2$ such that $A, \overline{A}, \overset{\circ}{A}, \overset{\circ}{\overline{A}}, \overline{\overset{\circ}{A}}$ are all different?
Can we get even more sets be alternating again closure and interior?

According to Kuratowski's closure-complement problem, the monoid generated by the complement operator $a$ and the closure operator $b$ has $14$ elements and is presented by the relations $a^2 = 1$, $b^2 = b$ and $(ba)^3b = bab$. Now you are interested by the submonoid generated by the closure operator $b$ and by the interior operator $i = aba$. This submonoid has only $7$ elements:
$1$, $b$, $i$, $bi$, $ib$, $bib$ and $ibi$. You can use Kuratowski's example
$$K = {]0,1[} \cup {]1,2[} \cup \{3\} \cup ([4,5] \cap \mathbb{Q})$$
to generate the $14$ sets and hence the $7$ sets you are interested in. This is an example in $\mathbb{R}$, but $K \times \mathbb{R}$ should work for $\mathbb{R}^2$.
