I'm not asking why it is an axiom of a topology that the void is open, but rather, when the topology of R^n is developped and open sets are defined as sets such that for any point in the set, we can find an epsilon-ball centered on that point that is entirely contained in the set. My book says that it follows from the dfn that the void is empty. How is that? If we argue that "since the void has no point, then it is true that for all points, we can find and epsilon-ball, etc.", then the opposite is just as true: "Since there are no point, we can say that for all point, we can never find an epsilon-ball, etc." There is no points in the void, so the definition simply does no apply it seems! Similar question: what's the boundary of the void? is it the void or the whole of R^n?