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In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. (1996), 1357–1372] about characterization of $\omega$-limit sets from the class $𝒞(I,I)$ of continuous maps of the interval to the class $𝒞(𝕊,𝕊)$ of continuous maps of the circle. Among others we give geometric characterization of $\omega$-limit sets and then we prove that the family of $\omega$-limit sets is closed with respect to the Hausdorff metric.

We extend the recent results from the class $𝒞(I,I)$ of continuous maps of the interval to the class $𝒞(𝕊,𝕊)$ of continuous maps of the circle. Among others, we give a characterization of $\omega$-limit sets and give a characterization of sets of transitive points for these maps.