Cantor Intersection Theorem extension Task at hand:
Show that in the Cantor Intersection Theorem, "compact" cannot be replased by "closed"; that is, find a nested sequence $\{F_n\}_{n=1}^\infty$ of nonempty closed sets in C such that $$\bigcap_{n=1}^\infty F_n=\varnothing $$
I know the difference between compact and closed but I am trying to find a nested sequence that meets the criteria but I can not seem to understand what is going on.
 A: We can take away boundedness from a closed set in $\mathbb R$ to make it non-compact.
$$\bigcap_{n}[n,\infty)=\varnothing$$
As $[n,\infty)$ is also closed in $\mathbb C$, this is a valid example for $\mathbb C$ also.
A: A family $F$ of sets has the F.I.P. (Finite Intersection Property) when every non-empty finite $G$ satisfies $\cap G\ne \phi.$ An equivalent def'n of compactness of a space is: A space $S$ is compact iff every non-empty family $F$ of closed sets with the F.I.P. satisfies $\cap F\ne \phi.$ So if $S$ is a non-compact space, it has a non-empty family $F$ of closed sets with the F.I.P. with $\cap F=\phi.$ In many cases such an $F$ may be $\{f_n: n\in N\}$, in which case we may let $g_n=\cap_{i=1}^nf_i$ and then $\{g_n :n\in N\}$ is a nested sequence of closed sets with empty common intersection. In a Hausdorff space, compact subsets are closed, and closed subsets of a compact set are compact sets. So in a Hausdorff space, if $G=\{g_n:n\in N\}$ is a nested family  of non-empty closed sets with $\cap G=\phi$ it will be necessary that none of the $g_n$ is compact.
