Any ideas on the second part? I am stuck.
Prove: a) Let be a transitive relation on the interval . If each has a neighborhood such that whenever and , then .
b) Use the above result to prove Cantors Nested Intervals Property (the intersection of a family of closed sets is non-empty).
A: Here is kind of an outline to part (a):
Let be arbitrary. Since is compact we know . Now, by our hypothesis, we can find a neightborhood of such that if and , then . So, there are two cases; first, is an interior point of the interval; second, is not an interior point, so a portion of the neighborhood may be outside the closed interval ( could even equal ). In either case, we can find a transitive relation [tex]\rho[/mathp such that implies , since we know [tex]b=sup([a,b]).
I am having troubles formalizing the proof and organizing my thoughts as you may be able tell.
b) I am stuck on this one even though my teacher offered the hint, "consider the relation iff and s.t. .