Does the property $\overline{C} = \overline{int(C)}$ hold for $C$, a closed subset of a normed space such that $int(C) \neq \emptyset$? I was thinking that perhaps this is false. For example, take the space $\mathbb{R}^2$ with the usual norm and topology. Then take $C$ to be a line and a closed ball (disjoint).
But then $\overline{int(C)}$ is just the closed ball...