In this paper, we introduce the selection principles wSS∗ 1(ΠΔ(Λ),CΔ(Λ)), wSS∗ fin(ΠΔ(Λ),CΔ(Λ)), wS∗ 1(ΠΔ(Λ),CΔ(Λ)) and wS∗ fin(ΠΔ(Λ),CΔ(Λ)) to characterize the properties of weakly strong-star Rothberger (Menger) and weakly star-Rothberger (Menger) in the hyperspace (Λ, τ+ Δ), respectively. Furthermore, we introduce the notions H(CΔ(Λ)) and Ifin(CΔ(Λ),CΔ(Λ)) to characterize, respectively, the H-separability and the principle Ufin(D,D), in the same hyperspace.