Show $A\subset (X,d)$ nowhere dense $\iff$ $(\overline{A})^o=\emptyset$.
My attempt: $A$ nowhere dense $\implies$ $(\overline{A})^c$ is dense in $X$. Then, $(A^c)^o$ is dense in $X$ (previously proven equivalence). Then, $\overline{((A)^c)^o}=X\implies\overline{((A)^c)^o}^c=\emptyset$. Then I have to show $\overline{((A)^c)^o}^c=(\overline{A})^o$. Am I going about this the right way?