Comparison of some set open topologies on \(C(X, Y)\)
type: Item
instance of: scholarly article
title: Comparison of some set open topologies on \(C(X, Y)\)
zbMATH Open document ID: 1305.54027
author: Abderrahmane Bouchair
author: S. Kelaiaia
published in: Topology and its Applications
publication date: 2014-11-14T00:00:00Z
review text: Let \(X\) and \(Y\) be two topological spaces and \(C(X, Y)\) be the set of all continuous functions from \(X\) into \(Y\). If \(\alpha\) is a nonempty family of subsets of \(X\), then the set open topology on \(C(X, Y)\) has a subbase consisting of the sets \([A, V] =\{f\in C(X, Y) :f(A) \subseteq V\},\) where \(A\in \alpha\) and \(V\) is an open subset of \(Y\). The function space \(C(X, Y)\) endowed with this topology is denoted by \(C_{\alpha}(X, Y).\) The set open topology was first introduced by \textit{R. Arens} and \textit{J. Dugundji} [Pac. J. Math. 1, 5--31 (1951; Zbl 0044.11801)]. In this paper the authors provide a criterion for the coincidence of the set open and uniform topologies on \(C(X, Y)\) for the case when \(Y\) is an equiconnected metric space. Also, they compare the spaces \(C_{\alpha}(X, Y)\) and \(C_{\beta}(X, Y)\) for two given families \(\alpha\) and \(\beta\) of compact subsets of \(X\) and study separation axioms on \(C(X,Y)\).
reviewed by: Dimitris Georgiou
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 6369803
zbMATH Keywords: function space
zbMATH Keywords: set open topology
zbMATH Keywords: topology of uniform convergence on a family of sets
MaRDI profile type: Publication
full work available at URL: j.topol.2014.10.008
OpenAlex ID: W2022641482
cites work: Topologies for function spaces
cites work: Q4150804
cites work: On fibre spaces. I
cites work: Q3756129
cites work: Topological properties of spaces of continuous functions
cites work: Some properties of set-open topologies
cites work: On the coincidence of set-open and uniform topologies
cites work: Some cardinal invariants on the space \(C_{\alpha}(X,Y)\)
DOI: 10.1016/J.TOPOL.2014.10.008
description: scientific article; zbMATH DE number 6369803