Let $A$ be a set in a metric space $M$.
Proof:
Assume $A^c$ is dense in M $\iff$ $B_{\epsilon}(x)\cap A^c $ for all $x \in M$ and for every $\epsilon > 0$ $\iff$ for all $x' \in A$ and every $\epsilon > 0$, $\ B_{\epsilon}(x') \cap A^c \neq \emptyset \iff$ there is no $B_{\epsilon}(x') \subset A$ for any $x' \in A$ and every $\epsilon > 0 \iff A^o$ is empty.
I am not sure if I can prove the equivalence in one go like this. I should note that I believe that $A^c = M \setminus A$ so it is trivial to show that for $x \notin A$ the epsilon ball around $x$ intersects $A^c$.
EDIT: $\iff$ for $x' \in A$ and $\epsilon > 0$ we cannot find $B_{\epsilon}(x') \subseteq A$.