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 ϕ : X → B ( Y ) , a locally bounded l.s.c. mapping Φ ( ϕ ) : X → B ( Y ) 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 ϕ : X → B ( Y ) , an l.s.c. mapping Φ ( ϕ ) : X → B ( Y ) and a u.s.c. mapping Ψ ( ϕ ) : X → B ( Y ) with ϕ ⊂ Φ ( ϕ ) ⊂ Ψ ( ϕ ) such that Φ ( ϕ ) ⊂ Φ ( ϕ ′ ) and Ψ ( ϕ ) ⊂ Ψ ( ϕ ′ ) whenever ϕ ⊂ ϕ ′ .