A theorem of Nepomnyashchii on continuous subset-selections (Q1888472)

Let \(X\) be a paracompact space, \(Y\) a completely metrizable one, and \(F: X\to Y\) a lower-semicontinuous multifunction with closed and equi-\(\text{LC}^0\) values. Let \(G: X\to Y\) be any upper-semicontinuous relation with compact (possibly empty) values such that \(G(x)\subset F(x)\), \(x\in X\). The following sandwich theorem is proved: if every \(F(x)\) is connected then there exists a continuous multifunction \(H: X\to Y\) with compact and connected values such that \(G(x)\subset H(x)\subset F(x)\), \(x\in X\). When \(G(x)= \emptyset\), \(x\in X\) this reduces to a multiselection theorem of Nepomnyashchii, although the proof is different. Four examples illustrating the essentiality of the assumptions are provided.