Let A be a set in the Euclidean space. Is the set A relative open with respect to its closure? Let $A$ be a connected set in the Euclidean space $\mathbb{R}^n$. Is the set $A$ relative open with respect to its closure $\bar{A}$?

Hint: Think of dense subsets (that is $\bar A = \mathbb R^n$). Are all such sets open?

For the altered question (connectedness added) you can find a counterexample as follows: Let 
$$ B = \bigl\{ (x, \sin x^{-1}) \mid x \in \mathbb R - \{0\} \bigr\} $$
Now let $A = B \cup \{(0,0)\}$. We have $\bar A = \bar B = B \cup (\{0\} \times [-1,1])$, and $(0,0)$ is not in the relative interior of $\bar A$.
