- #1
dimitri151
- 117
- 3
There the well known theorem that every open set (I'm talking about R here with standard topology) is the union of disjoint open intervals. Now, looking at the geometry, it seems that between any two adjacent open intervals which are in the union constituting our open set there is a closed interval. Therefore the union of all these disjoint closed sets that are between the open intervals constitute a closed set. Since every closed set is the complement of an open set, and every open set is the countable union of disjoint open intervals, every closed set is the countable union of disjoint closed intervals.
Any errors?
Any errors?