Weak factorizations of continuous set-valued mappings
type: Item
instance of: scholarly article
title: Weak factorizations of continuous set-valued mappings
zbMATH Open document ID: 1002.54012
DOI: 10.1016/S0166-8641(98)00139-4
published in: Topology and its Applications
publication date: 2000-06-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 54C50
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 1422595
zbMATH Keywords: multivalued map
zbMATH Keywords: selection
zbMATH Keywords: extension
zbMATH Keywords: hyperspace
Wikidata QID: Q114121884
author: Valentin Gutev
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1422595