Q:

Prove that the closure of complement, is the complement of the interior

Let $(X,d)$ a metric space and $A\subset X$. Prove that $$\overline{(A^c)}=\overset{\circ}{(A)}^c$$
i.e. the closure of complement, is  the complement of the interior. 
Proof.  If $x\in \overline{(A^c)}$ then $x\in A^c$ or $x\in (A^c)'$. Since $\overset{\circ}{A} \subset A \Rightarrow A^c \subset \overset{\circ}{(A)}^c$ so, if $x\in A^c$ then $x\in \overset{\circ}{(A)}^c$.
Now if $x \not\in A^c$ but $x\in (A^c)'$ then $\forall r>0, B(x;r)-\{x\}\cap A^c \not= \emptyset.$ And now i've to deduce that $x\not\in \overset{\circ}{A}$ so $x\in \overset{\circ}{(A)}^c$, but i can't figure out how. 

A:

If you use the topological definitions of closure and interior, it's very easy and natural:
$$\overline A=\bigcap\,\{F\supseteq A\mid F\text{ is closed in }X\}$$
$$A^\circ=\bigcup\,\{G\subseteq A\mid G\text{ is open in }X\}$$
Now just use the rules for complements which turn the complement of a union into an intersection of the complements. 

A:

I think this is easier to do if you consider the closure $\overline{A}$ as the set of elements of $X$ such that any open ball around them intersects $A$, that is, $$\overline{A} = \{ x \in X \mid \forall \epsilon > 0, B(x,\epsilon) \cap A \neq \emptyset \}.$$  Similarly, one definition of interior is the set of elements of $X$ such that you can find an $\epsilon$-ball around them contained in $A$, that is,$$A^{\circ} = \{ x \in X \mid \exists \epsilon > 0 \text{ s.t. }  B(x,\epsilon) \subseteq A \}.$$
Now, to prove the claim, first we will show $\overline{A^{c}} \subseteq (A^{\circ})^{c}:$

Let $x \in \overline{A^{c}}$.  Then for every $\epsilon > 0$, $B(x, \epsilon) \cap A^{c} \neq \emptyset$.  But this means any ball around $x$ will intersect with $A^{c}$, which means you can never find a ball around $x$ that is contained in $A$.   That means $x \not \in A^{\circ}$.  Thus, $x \not \in A^{\circ} \implies x \in (A^{\circ})^{c}$.

Now to show $(A^{\circ})^{c} \subseteq \overline{A^{c}}:$

Let $x \in (A^{\circ})^{c}$.  Then $x \not \in A^{\circ}$.  But this means there does not exist $\epsilon > 0$ such that $B(x,\epsilon) \subseteq A$.   But this means for every $\epsilon > 0$, $B(x,\epsilon) \cap A^{c} \neq \emptyset$, and this is precisely what we need for $x \in \overline{A^{c}}$.  So we are done. 

A:

Here is a nice 'natural language proof', based on the facts that the interior of a set is the largest open set contained in it, and the closure of a set is the smallest closed set that contains it.  By the way, this works for any topological space :
First, we note that $A^0$ is an open set contained in $A$, so $X\setminus A^0$ is a closed set that contains $X\setminus A$.  So we know that $X\setminus A^0\supseteq \overline{X\setminus A}$. Now, assume that $X\setminus A^0$ is not the smallest closed set containing $X\setminus A$.  Then, we would have some closed set $S$ such that $X\setminus A\subseteq S\subset X\setminus A^0$, which implies $A^0\subset X\setminus S\subseteq A$.  Since $X\setminus S$ is open, this means that $A^0$ is not the smallest open set containing $A$, which is a contradiction.