1. The problem statement, all variables and given/known data Show that if A ⊆ ℝ is both open and closed then A is either ℝ or ∅. 2. Relevant equations G ∩ bd(G) = ∅ ⇒ G is open bd(F) ⊆ F ⇒ F is closed bd(S) = bd(ℝ∖S) = bd(S') 3. The attempt at a solution Suppose A is a clopen set such that it is neither ℝ nor ∅ then ℝ∖A = A' is neither ℝ nor ∅. Now, A' is open because A is open (and closed). So ℝ = A ∪ A', where both A and A' are closed, which implies that ℝ is closed, a contradiction.