I'd like to help you with the first part.
One way to show that a set is closed is to show that it contains all of its limit points. So, take as in the problem, and put where is a noninterior point of .
Let be a limit point of . Then, there is a sequence of points from that converges to . If we can show that is an element of (i.e. a noninterior point of ), then is closed and we are done. To prove this, you should think about what happens if is not an element of ; in other words, what happens when is in the interior of ?