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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...

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  • $\begingroup$ I think, you are confusing "closed" and "compact". $\endgroup$ Dec 15 '14 at 16:31
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Take $X=\mathbb{R}$ and $A_m=[m,\infty)$.

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  • $\begingroup$ Is there an example if "closed" is replaced by "compact"? $\endgroup$
    – Nishant
    Dec 15 '14 at 16:36
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    $\begingroup$ 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 $\endgroup$
    – Miel Sharf
    Dec 15 '14 at 16:38
  • $\begingroup$ What's the proof of that? $\endgroup$
    – Nishant
    Dec 15 '14 at 18:28
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    $\begingroup$ 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. $\endgroup$
    – Miel Sharf
    Dec 15 '14 at 18:47