Show that A is nowhere dense
Given that $\displaystyle A\subset\overline{(X\setminus\overline{A})}.$
If $\displaystyle t\in A$ then $\displaystyle t$ is a point of $\displaystyle (X\setminus\overline{A})$ or a limit point of that set.
Find an open set, $\displaystyle \mathcal{O}$, containing $\displaystyle t$ but $\displaystyle \mathcal{O}\not\subset A~.$