There are different ways of defining "open" and "closed" set. The correct way of proving this depends on what definitions you are using.

One definition is that a set is "open" if it contains none of its "boundary" points and "closed if it containsallof its "boundary" points. If that is the definition you have then being "open" and "closed" means that "all" and "none" of its boundary points must be the same. What are the boundary points of the empty set? What about R? Can you show that any other set must have at least one boundary point?

Given a set S in Rn with the property that for any x that is an element of Sthere is an n-ball B(x) such that B(x)S is countable. Prove that S is countable.