Decompositions of set-valued mappings

Abstract

Let \(X\) be a set, \(B_{X}\) denotes the family of all subsets of \(X\) and \(F: X \to B_{X}\) be a set-valued mapping such that \(x \in F(x)\), \(\sup_{x\in X} | F(x)|< \kappa\), \(\sup_{x\in X} | F^{-1}(x)|< \kappa\) for all \(x\in X\) and some infinite cardinal \(\kappa\). Then there exists a family \(\mathcal{F}\) of bijective selectors of \(F\) such that \(|\mathcal{F}|<\kappa\) and \(F(x) = \{ f(x): f\in\mathcal{F}\}\) for each \(x\in X\). We apply this result to \(G\)-space representations of balleans.