Interior of Cluster of a set is a subset of its interior or not Let $(X,d)$ be any metric space. If we denote the cluster set of any $A\subseteq X$ with $\overline{A}$ and the interior set with $int(A)$,Is it true that $int(\overline{A})\subseteq int(A)$. If that is so how should I prove it, If not what is the counterexample. I have reached absolutely no counterexamples in the Euclidean space and somewhat needless to say in the discrete space. And I cannot proceed further than let $x\in int(\overline{A})$. Then by definition, for some $0<r_{0}\in\mathbb{R}$ the open neighborhood centered at x with radius $r_{0}$ is a subset of $\overline{A}$, that is $N_{r_{0}}(x)\subseteq\overline{A}$ and then I have not made any progress whatsoever. Any help is most appreciated.

Consider $\mathbb{R}$ with the usual euclidean metric, and $A = \mathbb{Q}$. Then $\mathrm{Int}(\bar A) = \mathbb{R} \not\subset \mathrm{Int}(A)=\varnothing$.
