1. The problem statement, all variables and given/known data A set E in ℝ is nowhere dense if and only if the closure of E's complement(E with a line over it) is dense in ℝ 2. Relevant equations I need help proving this lemma. I'm not entirely sure where to start. 3. The attempt at a solution I know we have to proof it two ways, backwards and forwards. For the forward, all I can think to use if the fact that the closure of E complement contains no nonempty open intervals, but I don't know where to go with that. Any help would be appreciated.