Let I1, I2, . . ., In be a ?nite set of closed intervals on the number line. So for instance, I1 might be [1, π], the set of all real numbers x such that 1 ≤ x ≤ π. The intervals are called closed because they contain their endpoints. Throughout this problem, suppose that every pair of intervals intersects.
a) Prove or disprove: all the intervals share a common point, that is, the intersection of all the intervals is nonempty.
b) Suppose the intervals need not be closed. They may be open, as in 1 < x < π, or half open, as in 1 ≤ x < π or 1 < x ≤ π. Prove or disprove: all the intervals share a common point.
c) Suppose the intervals are now line segments in the plane. Prove or disprove: all the intervals share a common point. (In this and the next part, the answer may depend on whether you restrict to closed intervals or not.)
d) Return to the number line, but now suppose there can be in?nitely many intervals. Again investigate: must all the intervals share a common point?