Lemma 4.5.8. Let $X$ be a topological space and $A \subset X$, then $(A^{o})^{c} = \overline{A^c}$.
Proof. Let $x \in X$. By (3) of Definition 4.5.1, $x \in (A^{o})^{c}$ if and only if there exists no $U \in \cn(x)$ such that $U \subset A$. Thus $x \in (A^{o})^{c}$ if and only if $U \cap A^{c} \ne \emptyset$ for all $U \in \cn(x)$. By (2) of Definition 4.5.2, this is equivalent to $x \in \ol{A^c}$.$\square$