1. The problem statement, all variables and given/known data Let R be the metric space of all real numbers. Prove that any bounded open set in R is a countable union of disjoint open intervals. 2. Relevant equations 3. The attempt at a solution If the bounded open set is continuous (is continuity defined for set?), then it is itself an open interval. So we let it be multiple continuous, intervals. Then each of the of those intervals is open, I think?