Let X be a compact metric space. By \(2^X\) we denote the hyperspace of all closed and non-empty subsets of X endowed with the Hausdorff metric. Let \(f:X\rightarrow X\) be a continuous function. We study some topological properties of the hyperspace \(\omega (f)\) , the collection of all omega limits sets \(\omega (x,f)\) with \(x\in X\) . We prove the following: (i) If X has no isolated points, then, for every continuous function \(f:X\rightarrow X\) , \({{\,\textrm{int}\,}}_{2^X}(\omega (f))=\varnothing \) . (ii) If X is a dendrite for which every arc contains a free arc and \(f:X\rightarrow X\) is transitive, then the hyperspace \(\omega (f)\) is totally disconnected. (iii) Let \(D_\infty \) be the Ważewski’s universal dendrite. Then there exists a transitive continuous function \(f:D_\infty \rightarrow D_\infty \) for which the hyperspace \(\omega (f)\) contains an arc; hence, \(\omega (f)\) is not totally disconnected.
