By [Cantor's intersection theorem](https://en.wikipedia.org/wiki/Cantor%27s_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?