I need to find a subset $A\subset \Bbb{R}$ such that the following sets are all different.
$$A\qquad \mathring{A} \qquad \overline{A}\qquad \overline{\mathring{A}}\qquad \mathring{\overline{A}}\qquad \mathring{\overline{\mathring{A}}}\qquad \overline{\mathring{\overline{A}}} $$
My best attempt was with the following set:
$A=(1,2)\cup (2,3)\cup \{4\}$
I have:
$$ \mathring{A}=(1,2)\cup(2,3) \\ \overline{A}=[1,3]\cup\{4\} \\ \overline{\mathring{A}}=[1,3] \\ \mathring{\overline{A}}=(1,3). $$
All of these sets are different, but
$$ \overline{\mathring{\overline{A}}}=\overline{\mathring{A}}=[1,3]\quad\textrm{and}\quad \mathring{\overline{\mathring{A}}}=\mathring{\overline{A}}=(1,3).$$
Can I add something to my attempt to fix this?