Decompositions of set-valued mappings

Abstract

Let X be a set, BX denotes the family of all subsets of X and F:X→BX be a set-valued mapping such that x∈F(x), supx∈X|F(x)|<κ, supx∈X|F−1(x)|<κ for all x∈X and some infinite cardinal κ. Then there exists a family F of bijective selectors of F such that |F|<κ and F(x)={f(x):f∈F} for each x∈X. We apply this result to G-space representations of balleans.