Semi-continuous multifunctions and bases of countable order
type: Item
instance of: scholarly article
title: Semi-continuous multifunctions and bases of countable order
zbMATH Open document ID: 0960.54013
DOI: 10.1016/S0166-8641(99)00015-2
author: Boualem Alleche
author: Jean Calbrix
published in: Topology and its Applications
publication date: 2001-05-14T00:00:00Z
review text: A base \({\mathcal B}\) for a topological space \(X\) is called a base of countable order, if for every \(x\in X\) and every strictly decreasing sequence \(U_n\) of elements of \({\mathcal B}\) containing \(x\), the \(U_n\) create a local base at \(x\). By a famous result of Arkhangel'skij, \(X\) is metrizable if and only if it is paracompact and has a base of such kind. Let \(X\) be a regular space possessing such a base \({\mathcal B}\) and \(Y\) be a paracompact space. It is proved that an lsc closed-valued multifunction \(F:Y\multimap X\) has an usc multiselector \(H:Y\multimap X\) and lsc multiselector \(G:Y\multimap X\), both compact-valued, such that \(G\subset H\subset F\), under the condition that \({\mathcal B}\) is monotonically complete on \(F(y)\) for every \(y\in Y\) (that means: for every decreasing sequence \(U_n\) of elements of \({\mathcal B}\) such that \(U_n\cap F(y) \neq\emptyset\) for every \(n=1,2,\dots\) the intersection of all closures of \(U_n\)'s is nonempty. An application concerning compact-covering maps is also presented. The main result is a substantial generalization of an important multiselector theorem of \textit{E. Michael} [Duke Math. J. 26, 647-651 (1959; Zbl 0151.30805)].
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E30
zbMATH DE Number: 1462689
zbMATH Keywords: multiselection
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: base of countable order
zbMATH Keywords: multifunction
zbMATH Keywords: multiselector
reviewed by: Włodzimierz Ślęzak
MaRDI profile type: Publication
full work available at URL: s0166-8641(99)00015-2
OpenAlex ID: W1982679611
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description: scientific article; zbMATH DE number 1462689