prove:
$\displaystyle \overline\emptyset = \emptyset$
$\displaystyle \overline X = X$
So you mean like $\displaystyle \overline A = A \cup A'$?
1) Suppose $\displaystyle x\in X$ were a "point of adherence" as you guys call them apparently. Then for any open U containing x, you need $\displaystyle U \cap \emptyset \not = \emptyset$ but this is clearly not possible. So the set of all points of adherence of $\displaystyle \emptyset$ is $\displaystyle \emptyset$.
2) Similarly suppose $\displaystyle x\in X$ were a "point of adherence." Then for any open U containing x, $\displaystyle U \cap X = U \supset \{x\} \not = \emptyset$. So every point in X is a point of adherence, so yeah, you are done.