1. The problem statement, all variables and given/known data Show that any two disjoint nonempty open sets are mutually separated. Show that any two disjoint nonempty closed sets are mutually separated. 2. Relevant equations 3. The attempt at a solution Let A and B be two disjoint nonempty open sets: Assume A and B share a boundary point. That boundary point exists in A intersect B. If the sets are disjoint, they can't intersect so this is a contradiction. Similar idea for closed sets. Is this correct?