Authors’ abstract: Let \({\mathcal K}\) be the class of compact subsets of \(I=[0,1]\), furnished with the Hausdorff metric. Let \(f\in C(I,I)\). The authors study the map \(\omega_ f: I\to {\mathcal K}\) defined as \(\omega_ f(x)=\omega (x,f)\), the \(\omega\)-limit set of \(x\) under \(f\). This map is rarely continuous, and is always in the second Baire class. Those \(f\) for which \(\omega_ f\) is in the first Baire class exhibit a form of nonchaos that allows scrambled sets but not positive entropy. This class of functions can be characterized as those which have no infinite \(\omega\)-limit sets with isolated points. The authors also discuss methods of constructing functions with zero topological entropy exhibiting infinite \(\omega\)-limit sets with various properties.