Boundaries of and selectors for upper semi-continuous multi-valued maps
type: Item
instance of: scholarly article
title: Boundaries of and selectors for upper semi-continuous multi-valued maps
zbMATH Open document ID: 0544.54016
DOI: 10.1007/BF01164155
author: J. E. Jayne
author: C. A. Rogers
author: R. W. Hansell
author: Iwo Labuda
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173583
review text: A multi-valued map F from a topological space X into a topological space Y is said to be upper semi-continuous (u.s.c.) if \(\{\) x:F(x)\(\subset U\}\) is open in X for each open set U in Y. The boundary of F at a point \(x_ 0\) in X is the subset of \(F(x_ 0)\) defined by \[ K(x_ 0)=F(x_ 0)\cap \cap_{V\in {\mathcal V}}cl[F(V)\backslash F(x_ 0)], \] where \({\mathcal V}\) is the set of all neighbourhoods of \(x_ 0\). It is proved that if \(x_ 0\) is a first countable point in a Hausdorff space X and F is an u.s.c. map of X into an angelic space Y, then \(K(x_ 0)\) is compact; if \(x_ 0\) is a q-point (in the sense of Michael) in a regular Hausdorff space X, F is an u.s.c. map of X into a Dieudonné complete space Y, and each point of \(Y\backslash F(x_ 0)\) is contained in a \({\mathcal G}_{\delta}\)-set disjoint from \(F(x_ 0)\), then \(K(x_ 0)\) is compact. These compactness results are used to extend earlier measurable selection results of Jayne and Rogers. It is proved that if F is an u.s.c. map, with arbitrary non-empty values, from a metric space X into the Banach space \(c_ 0(\Gamma)\) with its weak topology, then F has a selector f of the second Borel class using the norm topology of \(c_ 0(\Gamma)\); if F is an u.s.c. map, with arbitrary non-empty values, from a metric space X into a Banach space Y with its weak topology, and if Y is K-analytic in its weak topology, then F has a Baire measurable selector, using the weak topology on Y.
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 3866162
zbMATH Keywords: first countable point
zbMATH Keywords: Hausdorff space
zbMATH Keywords: angelic space
zbMATH Keywords: Dieudonné complete space
zbMATH Keywords: Baire measurable selector
MaRDI profile type: Publication
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OpenAlex ID: W1990873737
description: scientific article; zbMATH DE number 3866162