1

What is an example of a complete metric space $X$ and a nested sequence of closed sets $A_m \subset X$ such that $\bigcap_{n=1}^\infty A_m = \emptyset$? My analysis professor mentioned this in office hours as something interesting to think about, and I've thought about it for a while. But I haven't made much progress...

Simon S
  • 26,524

1 Answers1

5

Take $X=\mathbb{R}$ and $A_m=[m,\infty)$.

Miel Sharf
  • 525
  • Is there an example if "closed" is replaced by "compact"? – Nishant Dec 15 '14 at 16:36
  • 1
    I think that the answer is no - All of the subsets are contained in $A_1$, which would be compact, and in compact spaces one has the finite intersection property: if there is a collection of subsets ${A_i}{i\in I}$ such that the intersection of finitely many of them is always nonempty, then $\bigcap{i\in I}{A_i}$ is non-empty – Miel Sharf Dec 15 '14 at 16:38
  • What's the proof of that? – Nishant Dec 15 '14 at 18:28
  • 1
    Take complements of everything - you conclude that if there is no finite subcollection of the open sets ${A_i^c}_i$ which covers the whole space, then the entire collection is not a cover, i.e. every open cover must have a finite subcover. – Miel Sharf Dec 15 '14 at 18:47