On the greatest splitting topology
type: Item
instance of: scholarly article
title: On the greatest splitting topology
zbMATH Open document ID: 1157.54005
published in: Topology and its Applications
publication date: 2008-12-05T00:00:00Z
review text: Let \(Y,Z\) be topological spaces and let \(C(Y,Z)\) denote the set of continuous mappings from \(Y\) into \(Z\). It is well known that the intersection of all admissible topologies on \(C(Y,Z)\) is the greatest splitting topology \(t\). If \(Y\) is regular and locally compact, then \(t\) is the compact open topology which is also admissible. Let \(Y\) have topology \(\tau\) and define the semi-regular reduction of \(Y\) to be the topology \(\tau_{sr}\) with base \(\{\text{int(Cl}(U)):U \in\tau\}\). We say that two spaces \((Y,\tau^0)\) and \((Y,\tau^1)\) are sr-reduction equivalent in case \(\tau^0_{sr}=\tau^1_{sr}\). In the present paper the authors show that if \(Y\) is sr-reduction equivalent to a regular locally compact space, and \(Z\) is regular, then the greatest splitting topology on \(C(Y,Z)\) is admissible. They then show how these hypotheses can be modified in various ways to obtain further results. A number of counter-examples and open questions are also given.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 5378343
zbMATH Keywords: splitting topology
zbMATH Keywords: admissible topology
zbMATH Keywords: greatest splitting topology
zbMATH Keywords: semiregularity
author: Dimitris Georgiou
author: S. D. Iliadis
MaRDI profile type: Publication
full work available at URL: j.topol.2007.11.008
OpenAlex ID: W1986526115
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DOI: 10.1016/J.TOPOL.2007.11.008
description: scientific article; zbMATH DE number 5378343