1. The problem statement, all variables and given/known data Give an example of a nonempty set A in R such that A = Bd(A) = Lim(A) = Cl(A). Bd(A) is the boundary of A, Lim(A) is the set of limit points of A, Cl(A) is the closure of A. 2. Relevant equations Bd(A) = A - Int(A), Int(A) is the interior of A Lim(A) = is the set of all such points in R, such that any size neighborhood around any of those points contains at least one point of A different from the point itself. Cl(A) = Lim(A)U(A) 3. The attempt at a solution Since Bd(A) = A - Int(A), then for A = Bd(A) => Int(A) is empty for A = Lim(A), not only are we looking for a closed set (due to it containing its boundary points), but one which is uncountable. For example, if A = integers, then none of the points in A are in Lim(A), in fact Lim(A) becomes empty. This is what makes me think that in between any 2 points in A, there must always be another point of A. And since Cl(A) = Lim(A) U (A), it will instantly follow once we get Lim(A) = A. Some of my attempts have been the singleton set, which makes Lim(A) empty. Then I tried the irrationals, but since they're dense, any rational number would become a limit point of A, thus making Lim(A) = R. This leads me to also restrict A to not be dense. Such would also rule out the set of rational numbers. After this, I can't deduce any other necessary and therefore neither any sufficient properties for A. I would appreciate a couple hints and/or pointing out any error(s) in my reasoning. Thank you for your time.