Topologies on the space of continuous functions
type: Item
instance of: scholarly article
title: Topologies on the space of continuous functions
zbMATH Open document ID: 0940.54023
DOI: 10.1016/S0166-8641(97)00114-4
author: V. Pereyra
published in: Topology and its Applications
publication date: 1999-10-11T00:00:00Z
review text: The authors study the coincidence of the fine topology \(\tau_{\omega}\) (Whitney topology, Morse topology, \(m\)-topology) on the set \(C(X,Y)\) of all continuous functions from a topological to a metric space with some other function space topologies. (Note that the fine topology depends on the concrete metric on \(Y\), not only on the underlying topology, whereas some of the compared topologies are completely topologically defined.) They show that \(\tau_{\omega}\) coincides with the graph topology, if \(X\) is countably paracompact and normal, with the Krikorian topology, if in addition the metric space \(Y\) is separable (moreover, it is shown, that the Krikorian topology coincides with the graph topology, if \(Y\) is a topological space with countable base and \(X\) is countably paracompact and normal), with the topology of uniform convergence and the open-cover topology, if \(X\) is countably compact and \(Y\) paracompact and first countable. Using cardinal functions, some interesting lemmas concerning the fine topology are established. The equivalence of first countability, metrizability, complete metrizability, Čech-completeness and pseudocompactness is shown for \((C(X,IR),\tau_{\omega})\) without any requirements on \(X\). Furthermore, the space \(W\) of all ordinals less than the first uncountable ordinal is examined and the coincidence of uniform convergence, Krikorian, graph and open-cover topology on \(C(W,W)\) is shown.
reviewed by: Harry Poppe
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 1307975
zbMATH Keywords: fine topology
zbMATH Keywords: m-topology
zbMATH Keywords: Krikorian topology
zbMATH Keywords: open-cover topology
zbMATH Keywords: graph topology
zbMATH Keywords: topology of uniform convergence
zbMATH Keywords: pseudocompact
zbMATH Keywords: countably compact
zbMATH Keywords: countably paracompact
zbMATH Keywords: equinormal
zbMATH Keywords: cardinal functions
zbMATH Keywords: function space
zbMATH Keywords: coincidence of topologies
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1307975