On a family of \(\lambda\)-topologies on a function space
type: Item
instance of: scholarly article
title: On a family of \(\lambda\)-topologies on a function space
zbMATH Open document ID: 0949.46010
DOI: 10.1007/BF02673897
published in: Siberian Mathematical Journal
publication date: 2000-08-30T00:00:00Z
review text: The author deals with the space \(C(X)\) of all continuous functions defined on a totally regular space \(X\). A set \(A\subset X\) is called bounded if every function in \(C(X)\) is bounded on \(A\). Each family \(\lambda\) of bounded subsets of \(X\) such that \(\widetilde\lambda =\cup\lambda\) is dense in \(X\) determines the so-called \(\lambda\)-topology on \(C(X)\) which is the topology of uniform convergence on the elements of \(\lambda\). The family \(\Lambda T(X)\) of all \(\lambda\)-topologies splits into the classes \(\Lambda_s T(X)=\{T_{\lambda}: \widetilde\lambda =s\}\) over dense subsets \(s\) of \(X\). Each family \(\mathcal A\subset \Lambda T(X)\) generates the topologies \(\inf\mathcal A\) and \(\sup\mathcal A\), the greatest lower and least upper bounds of \(\mathcal A\) in the complete lattice of all locally convex topologies on \(C(X)\). The main results are as follows: (1) the family \(\sigma\Lambda T(X)\) of all weak topologies generated by \(\Lambda T(X)\) does not coincide with \(\Lambda T(X)\) and the intersection of these families contains only the topologies of pointwise convergence; (2) the same holds for the family \(\inf\Lambda T(X)\); (3) \(\inf\Lambda_s T(X)=\Lambda_s T(X)\) if and only if every closed bounded subset of \(X\) is compact; (4) the operations \(\sigma\) and \(\inf\), as well as \(\sigma\) and \(\sup\), commute (\(\sigma\inf =\inf\sigma\) and \(\sigma\sup =\sup\sigma\)).
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 1498405
zbMATH Keywords: totally regular space
zbMATH Keywords: locally convex topology
zbMATH Keywords: weak topology
zbMATH Keywords: topology of uniform convergence
zbMATH Keywords: \(\lambda\)-topology
author: Nikolay V. Velichko
MaRDI profile type: Publication
cites work: Q5613420
cites work: Espaces de fonctions continues
full work available at URL: https://doi.org/10.1007/bf02673897
OpenAlex ID: W1987233323
description: scientific article; zbMATH DE number 1498405