Exercise 3.5.8

First suppose $E$ is nowhere-dense, then $\bar{E}$ contains no nonempty open intervals meaning for every $a,b\in R$ we have $(a,b)⊈\bar{E}$ meaning we can find a $c\in (a,b)$ with $c\notin \bar{E}$. But this is just saying $c\in {\bar{E}}^c$ which implies ${\bar{E}}^c$ is dense since for every $a,b\in R$ we can find a $c\in {\bar{E}}^c$ with $a<c<b$.

Now suppose ${\bar{E}}^c$ is dense in $R$, then then every interval $(a,b)$ contains a point $c\in {\bar{E}}^c$, implying that $(a,b)⊈\bar{E}$ since $c\notin \bar{E}$ and $c\in (a,b)$. therefore $\bar{E}$ contains no nonempty open intervals and so $E$ is nowhere-dense by defintion 3.5.3.