Abstract
We introduce the notion of πF(Δ)–network and denote the combinatorial principles FELL(ΠF(Δ),ΠF(Δ)) and FELL⁎(ΠF(Δ),ΠF(Δ)), which will be applied to characterize the spaces X whose hyperspace, endowed with the Fell topology, satisfies the SSR condition and the SR condition. We use the selection principle SSΔ⁎(O,O) to characterize the SSR property for the spaces K(X), F(X) and [X]1, endowed with the lower Vietoris topology. In addition, we introduce the selection principle SΔf(O,O) to obtain characterizations of the selective versions of metacompactness, mesocompactness and sequential mesocompactness. Finally, we introduce the notion of Δ-moving-off family, which generalizes the one of moving-off family, and we use it to characterize the Rothberger property for certain subspaces of CL(X).
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