We'll need your definitions, for an open set may be defined this way. What is a boundary point? Open set? The usual definition of an open set U is that every point x in U has an open set N_x with N_x strictly contained in U. If U is open and if a boundary point y lies in U, then every open set around y contains infinitely many points not in U. Conversely, if a set U contains no boundary points and U is not open then there is some point x in U that does not have a neighborhood strictly contained in U. Which means x is a boundary point.
