Let $(X, d)$ be a metric space with no isolated points, and let $A$ be a relatively discrete subset of $X$. Prove that $A$ is nowhere dense in $X$.
relatively discrete subset of $X$:= A subset $A$ of a topological space $(X,\mathscr T)$ is relatively discrete provided that for each $a\in A$, there exists $U\in \mathscr T$ such that $U \cap A=\{a\}$.
My aim is to prove $int(\overline{A})=\emptyset$. Let if possible $int(\overline{A})\neq \emptyset$. Let $x\in int(\overline{A})$. which implies there exists $B_d(x,\epsilon)\subset \overline A=A\cup A'$.
How do I complete the proof?
What if metric space is replaced by arbitrary topological space, will the result still hold?