Reference no: EM13588352

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?