Show $(A^o)^c=\overline{A^c}$.
($\rightarrow$) $(A^o)^c\subseteq\overline{A^c}$
I want to show that $(A^o)^c$ is closed and that $A^c\subseteq (A^o)^c$. Then ($\rightarrow$) follows. Since $A^o$ is open $(A^o)^c$ is closed. Since $A^o\subseteq A$, $A^c\subseteq (A^o)^c$. Done.
($\leftarrow$) $\overline{A^c}\subseteq (A^o)^c$
This one is trickier. $(A^o)^c$ is closed and since $A^o\subseteq A\implies$ $A^c\subseteq (A^o)^c$. So $\overline{A^c}\subseteq (A^o)^c$ since $\overline{A^c}$ is the smallest closed set which $A^c$ fits into?