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In proving Cantor's in intersection theorem, the fact that limit of the diameter of the sets is 0 was used to prove that the intersection is non-empty. I just wondered if that hypothesis is excluded can the intersection be empty?

Specifically in a compact metric space can there be a sequence of non-empty nested closed sets such that the intersection is empty.

amWhy
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Infinity
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    No, the diameter limit zero is to guarantee no more than one point in the intersection. Together with nonempty that gives intersection is a unique point. – coffeemath Oct 23 '14 at 12:16

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In a non-compact space, sure. Just consider closed rays on the real line, e.g. $[n, \infty)$ for $n$ a natural number.

In a compact space a family of closed sets that every finitely many have a non-empty intersection, will have a non-empty intersection as well. So the diameter is not needed in showing that the intersection is non-empty.

Asaf Karagila
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    On the other hand, the intersection theorem with diameters going to zero works in complete metric space, even one that is not compact. – GEdgar Oct 23 '14 at 12:31