False: if $C$ is closed then closure of interior of $C$ is equal to $C$? If $C$ is a closed set in a metric space $(X,d)$, then $\overline{C^\circ} = C$
I know that this is false, but I'm having trouble coming up with a good
counterexample to show that it doesn't work. Ideas?
Edit:
Wow, the answers are so simple! Major brain fart... Thank you so much for input!

$C=[0,1]\cup \{2\}$ is closed. $C^o=(0,1)$. $\overline{C^o}=[0,1]\ne C$
