In Problems 1 to 6, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample.
1. If the intersection of two open intervals is nonempty, then their intersection is an open interval.
2. If the intersection of two closed intervals is nonempty, then their intersection is a closed interval.
3. The union of any two open intervals is an open interval.
4. The union of any two closed intervals is a closed interval.
5. If the intersection of two open intervals is nonempty, then their union is an open interval
6. If the intersection of two closed intervals is nonempty, then their union is a closed interval.