Let \(\omega_f\) denote the class of all \(\omega\)-limit sets of a continuous self-mapping of a real compact interval \(I\). If \(W\in \omega_f\), then \(W\) is a closed nonempty subset of \(I\) and \(f(W)=W\). Consider \(\omega_f\) as a subspace of the compact metric space \(K\) of all closed nonempty subsets of \(I\) furnished with the Hausdorff metric. It is shown that \(\omega_f\) is a closed and therefore compact subspace of \(K\). Results are then applied to other dynamical systems.