Let $A$ be a subset of a metric space $X$ and let $x_o$ be an isolated point of $A$. Show $x_o\in\partial A$ iff $x_o\in Acc(A^{c})$.
My attempt:
($\rightarrow$) Let $x_o\in\partial A$. Then, $B_\epsilon(x_o)\bigcap A \neq \emptyset$ and $B_\epsilon(x_o)\bigcap A^{c} \neq \emptyset$, $\forall\epsilon>0$. Using the latter fact, we see $x_o\in Acc(A^{c})$.
($\leftarrow$) Let $x_o\in Acc(A^{c})$. Then, $\forall\epsilon>0,B_\epsilon(x_o)\bigcap A^{c} \neq \emptyset$ Since $x_o$ is an isolated pt. of A, $\exists\epsilon>0$ s.t. $B_\epsilon(x_o)\bigcap A={x_o}\neq\emptyset$. Thus $x_o\in\partial A$.