If $A$ is a closed set with $\mathring{A}=\emptyset$, it is the border of another set I have to decide whether the following statement is true:

If $A\neq\emptyset$ is a closed set, and $\mathring{A}=\emptyset$, then $\exists D$ so that $A=\partial D$, where $A, D \in X$

I know that if $X=\mathbb{R^2}$ (for example) this is true and it's easy to proveit, since I just have to take $D$ as the rational elements in $A$. However, for any set $X$ this method doesn't work. Can someone help me with this?

Yes, it is true since $$\partial A = \overline{A} \setminus \overset{o}{A} =\overline{A} \setminus \emptyset =A.$$
