The hyperspace ω(f) when f is a transitive dendrite mapping

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 $f:X\to X$ be a continuous function. In this paper 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 $f:X\to X$, $in{t}_{2^X}(\omega (f))=\mathrm{\varnothing }$. $ii)$ If $X$ is a dendrite for which every arc contains a free arc and $f:X\to X$ is transitive, then the hyperspace $\omega (f)$ is totally disconnected. $iii)$ Let $D_{\mathrm{\infty }}$ be the Wazewski’s universal dendrite. Then there exists a transitive continuous function $f:D_{\mathrm{\infty }}\to D_{\mathrm{\infty }}$ for which the hyperspace $\omega (f)$ contains an arc; hence, $\omega (f)$ is not totally disconnected.