USCO and quasicontinuous mappings
type: Item
instance of: scholarly article
title: USCO and quasicontinuous mappings
zbMATH Open document ID: 1492.54001
DOI: 10.1515/9783110750188
author: Dušan Holý
author: Ľubica Holá
author: Warren B. Moors
published in: De Gruyter Studies in Mathematics
publication date: 2021-12-02T00:00:00Z
review text: In the preface of this book the following goal is set by the authors, who are noted experts on the topic: ``The purpose of our monograph is: \begin{itemize} \item[--] to introduce to nonspecialists the classes of usco and quasicontinuous mappings, \item[--] to present the most fundamental properties of these mappings, \item[--] to highlight their utility.'' \end{itemize} A first question arises: what is the main idea which led the authors to select, among various generalizations of continuity, in particular quasicontinuity and upper semicontinuity, and to discuss both together in the same book? Although these two notions are closely related via the continuity of single-valued maps, they are also, at first glance, rather distinct generalizations: separately quasicontinuous functions on product spaces are jointly quasicontinuous (and thus behave even better than continuous functions) whereas separately upper semicontinuous functions or multifunctions may even fail to be measurable (and thus behave worse); the sum of two (real) upper semicontinuous functions is usc (behaves like continuous functions) whereas the sum of two quasicontinuous functions may fail to be quasicontinuous (in fact each real cliquish function is the sum of two quasicontinuous functions). There are numerous answers to this first question, but each such answer assumes the existence of some common connecting leitmotif, forcing the subordinated structure of the book (the graph of content dependence should minimize the number of its leaf vertices). In the case of this book (the first monograph dedicated to this subject in the literature) the common idea connecting quasicontinuity and usco-ness are common applications, including such interesting areas as semitopological or paratopological groups, Glicksberg's theorem and others. The deepest interrelations between both generalizations of continuity are contained in profound interactions between usco multifunctions and their quasicontinuous selections (T…
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 49J53
zbMATH DE Number: 7438191
zbMATH Keywords: upper semicontinuous multifunction
zbMATH Keywords: quasicontinuous selection
zbMATH Keywords: minimal usco mappings
zbMATH Keywords: subdifferential
zbMATH Keywords: differentiability of Lipschitz functions
zbMATH Keywords: cusco mappings
reviewed by: Włodzimierz Ślęzak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110750188
OpenAlex ID: W3205270695
description: scientific article; zbMATH DE number 7438191