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The space of $$\omega$$-limit sets of a continuous map of the interval. (English) Zbl 0860.54036
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.

##### MSC:
 54H20 Topological dynamics (MSC2010) 26A18 Iteration of real functions in one variable 37E99 Low-dimensional dynamical systems 37B99 Topological dynamics
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