of a countable collection of open intervals. I'm having a hard time seeing how this could be true. For instance, take the open set (0, 10). I'm having a hard time seeing how one could make this into a union of countable open intervals. For instance, (0,1) U (1, 10) or (0, 3) U (3, 6) U (6, 10) wouldn't work because those open intervals miss some points. There are "gaps" missing from the initial open set (0, 10). It seems like any union of DISJOINT intervals would have "gaps" missing from the initial open set. And if any of the open sets overlap to fill those gaps, then they are no longer disjoint. I've read several proofs of this theorem, and they don't clear up my confusion.