Colorings for set-valued maps (Q2818190)

For a space \(X\) denote by \(2^X\) the space of all non-empty closed subsets of \(X\) endowed with the Vietoris topology. Let \(X\subset Y\) be spaces and \(f:X\to2^Y\) a map. A subset \(A\) of \(X\) is called a colour if \(A\cap f(A)=\emptyset\). A colouring of \(f\) is a finite closed cover of \(X\) consisting of colours of \(f\). In this case \(f\) is said to be colourable. Denote by \(\mathcal{F}_k\) the subspace of \(2^X\) consisting of sets with at most \(k\) elements. The authors prove the following theorem: Let \(X\) be a paracompact space with \(\dim X\leq n\) and let \(f:X\to\mathcal{F}_k\) be a fixed-point free upper semicontinuous map such that \(l:=\sup\{\#f^{-1}(\{x\})|\;x\in X\}<\infty\). Assume that \(f\) maps closed sets into closed sets. Then \(f\) is colourable with at most \((k+l)(n+1)+1\) colours. There is a similar result in case \(X\) is separable metric.