Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa Give an example of a metric space $(X,d)$ and  $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and $\overline{\text{int}(A)}\not\subseteq\text{int}(\overline{A})$.
I've tried finding an appropriate interval in $(\mathbb{R}, d_{\text{eucl}})$ but I've been unsuccessful. I've also experimented with the discrete metric, but as every subset is both open and closed, the interior of the closure always ends up being equal to the closure of the interior. I'm having a difficult time visualizing the concept of open/closed sets in other metric spaces.

Consider $\Bbb R$ with the usual metric.  Let $A$ be the set of rationals from $0$ to $1$ together with the set of reals from $2$ to $3$, all four endpoints excluded.  Then
$${\rm int}(\overline A)=(0,1)\cup(2,3)$$
and
$$\overline{{\rm int}(A)}=[2,3]\ .$$
Neither is a subset of the other.
