Continuous convergence of mappings
type: Item
instance of: scholarly article
title: Continuous convergence of mappings
zbMATH Open document ID: 0840.54034
published in: Russian Mathematics
publication date: 1996-01-03T00:00:00Z
review text: The author uses Robinson-styled nonstandard analysis to study properties of the set of all multivalued functions, \({\mathcal F}\), considered as subsets of \(X\times Y\), where usually \(X\) and \(Y\) are pre-topological convergence spaces. The author's major interest is relative to the concepts of upper (UC) and lower (LC) continuity. Such a relation is said to be continuous (C) if it is both upper and lower continuous. These three concepts are defined in terms of the monad generating maps \(\rho\) and \(\sigma\) considered as functions defined on \(X\times {}^* X\), \(Y\times {}^* Y\), respectively and represented as subsets of such products. For pre-topological or topological spaces, these are the usual monads associated with such spaces. For example, \(f\in {\mathcal F}\) is upper continuous (i.e. \(f\in {\mathcal F}_{\text{UC}}\)) if \({}^* f\circ\rho \subset \sigma\circ f\). I point out that when such functions \(f\) are single-valued, then some of the author's results overlap with known results relative to the nonstandard approach to pseudo- and pre-topological convergence spaces. The major results are in terms of convergence type structures defined on \({\mathcal F}\). For example, if \((f, h)\in {\mathcal F}\times {\mathcal F}\), then \((f,h)\in \gamma_{\text{UC}}\) iff \(h\circ \rho\subset \sigma\circ f\) and, in this case, is called an upper continuous convergence on \({\mathcal F}\). Let \(\lambda: {\mathcal F}\times X\) be defined for each \(x\) in the domain of \(f\) by letting \(\lambda (f,x)= f(x)\). Then as an example of these concepts, it is established that continuous convergence \(\gamma_{\text{UC}}\) is the weakest convergence of \({\mathcal F}_{\text{UC}}\) for which \(\lambda\) is continuous in the corresponding sense. The author establishes many other interesting and useful results.
Mathematics Subject Classification ID: 54J05
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 815635
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: convergence spaces
author: V. A. Molchanov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 815635