Closure of interior proper subset of interior of closure

Question

$\newcommand{\intr}{\mathrm{int}}$

I need to give an example of a metric space $(X,d)$ and $A ⊆ X$ so that $\overline{\intr(A)} ⊂ \intr (\overline{A})$, where $\overline{B}$ refers to the closure of $B$.

Also, another $X,A$ where there is no containment relation beween the two.

I've found simple examples for $\overline{\intr(A)} ⊃ \intr(\overline{A})$ and for equality, but I'm guessing I need to look further afield here.

I'd appreicate any hints.

Answer 1

With the usual metric on $\mathbb{R}$, Let $ X=\mathbb{R}$ and $A= \mathbb{Q}$.