Locally bounded set-valued mappings and monotone countable paracompactness

We prove that the following statements are equivalent for a space X: (1) X is monotonically countably paracompact; (2) for every metric space Y there exists an operator Φ assigning to each locally bounded mapping , a locally bounded l.s.c. mapping with ϕ⊂Φ(ϕ) such that Φ(ϕ)⊂Φ(ϕ′) whenever ϕ⊂ϕ′, where B(Y) is the set of all non-empty closed bounded sets of Y; (3) for every metric space Y, there exist operators Φ and Ψ assigning to each u.s.c. mapping , an l.s.c. mapping and a u.s.c. mapping with ϕ⊂Φ(ϕ)⊂Ψ(ϕ) such that Φ(ϕ)⊂Φ(ϕ′) and Ψ(ϕ)⊂Ψ(ϕ′) whenever ϕ⊂ϕ′.