Weak factorizations of continuous set-valued mappings (Q1971364)

The main result. Let \(X\) be a topological space, \(Y\) a completely metrizable space and \(\mathcal{F}(Y)\) denote the collection of all nonempty closed subsets of \(Y\). Let \(\Phi : X \rightarrow \mathcal{F}(Y)\) be a proximal continuous multimap, i.e. it is both upper semicontinuous with respect to a metric of \(Y\) and lower semicontinuous. Further, let \(A \subset X\) and \(g: A \rightarrow Y\) be a selection for \(\Phi_{|A}\) such that for every locally finite cozero-set cover \(\mathcal{V}\) of \(Y\) there exists a locally finite cozero-set cover \(\mathcal{U}\) of \(X\) such that \(g(\mathcal{U} \cap A)\) refines \(\mathcal{V}\). Then the multimap \(\Phi_g\) defined as \(\Phi_g(x) = \{g(x)\}\) if \(x \in A\) and \(\Phi_g(x) = \Phi(x)\) otherwise, admits an l.s.c. weak factorization, i.e. there exist a metrizable space \(Z\) with weight \(w(Z) \leq w(Y)\), a continuous map \(h: X \rightarrow Z\) and an l.s.c. multimap \(\phi: Z \rightarrow \mathcal{F}(Y)\) such that \(\phi \circ h\) is a selection of \(\Phi\). This result is applied to obtain generalizations of some known selection and extension results and to selection theorems for spaces of closed subsets.