TITLE: How is compactness related to countable saturation?
QUESTION [6 upvotes]: By Cantor's intersection theorem every decreasing nested sequence of nonempty compact sets has a common point. 
A superficially similar result holds that every decreasing nested sequence of nonempty internal sets in an ultrapower model of a hyperreal field ${}^\ast\mathbb{R}$ has a common point, a property known as countable saturation. 
Is such a resemblance more than superficial?

REPLY [2 votes]: It turns out that there is a direct relationship between Cantor's lemma for nested compact sets $K_n$, on the one hand, and nested sequences of internal sets, on the other. Namely, the latter can be viewed as a special case of the former as follows.
The decreasing nested sequence of internal sets, $\langle{}^\ast\!K_n\colon n\in\mathbb N\rangle$, has a common point $x$ by saturation. But for a compact set $K_n$, every point of ${}^\ast\!K_n$ is nearstandard, i.e., infinitely close to a point of $K_n$ (in other words, every point is in the halo of a standard point of $K_n$). In particular, $st(x)\in K_n$ for all $n$, as required.