Generalized metric properties and kernels of set-valued maps
type: Item
instance of: scholarly article
title: Generalized metric properties and kernels of set-valued maps
zbMATH Open document ID: 1071.54012
author: Jiling Cao
published in: Topology and its Applications
publication date: 2005-02-22T00:00:00Z
review text: Let \(F: X\to Y\) be a multifunction between topological spaces \(X\) and \(Y\). A subset \(K\subset X\) is called a kernel of \(F\) at \(x\in X\) if for each open set \(V\supset K\) there is a neighbourhood \(N\ni x\) with the property \(F(N)\subset V\cup F(x)\). The Choquet-Dolecki theorem states that if \(X\) is first-countable and \(Y\) is metrizable, then any upper semicontinuous \(F\) has a compact kernel at every \(x\in X\). The present paper proves that this theorem remains true if \(X\) is a \(\mathcal G\)-space and \(Y\) a \(k\)-semistratifiable one. Topological game theory is heavily used in the proof.
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 26E25
zbMATH DE Number: 2136494
zbMATH Keywords: upper semicontinuous multifunction
zbMATH Keywords: kernel
zbMATH Keywords: stratifiable space
zbMATH Keywords: \(g\)-space
zbMATH Keywords: topological game
reviewed by: Włodzimierz Ślęzak
MaRDI profile type: Publication
full work available at URL: j.topol.2004.03.003
OpenAlex ID: W2071765655
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cites work: Semimetrizable and stratifiable spaces
DOI: 10.1016/J.TOPOL.2004.03.003
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description: scientific article; zbMATH DE number 2136494