Boundary-type sets have empty interior and locally finite unions commute with closure

Let \(A\) be a subset of a topological space \(X\). Prove that both \[ \overline A\setminus A \qquad\text{and}\qquad A\setminus\operatorname{int}A \] have empty interior.

A family \(\mathcal A\) of subsets of a topological space \(X\) is locally finite if for every \(x\in X\) there exists a neighborhood \(U\) of \(x\) such that \(U\) meets only finitely many members of \(\mathcal A\). If \(\mathcal A\) is locally finite, prove that \[ \overline{\bigcup_{A\in\mathcal A}A} = \bigcup_{A\in\mathcal A}\overline A. \]