The compact-open topology for semigroups of continuous selfmaps
type: Item
instance of: scholarly article
title: The compact-open topology for semigroups of continuous selfmaps
zbMATH Open document ID: 0608.22004
DOI: 10.1007/BF02573088
author: Saraswathi Subbiah
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134854
review text: All spaces in this discussion are assumed to be Hausdorff. Let \(S(X)\), the semigroup of all continuous selfmaps of the topological space \(X\) have the compact-open topology. It is well known that \(S(X)\) is a topological semigroup if \(X\) is locally compact. It does not seem to be known that the converse is ``almost true'' in the sense that there is an extensive class of spaces with the property that for any such space \(X\), if \(S(X)\), with the compact-open topology, is a topological semigroup, then \(X\) must necessarily be locally compact. In the main result, we establish this fact and we discuss an example which shows that there do exist spaces \(X\) such that \(S(X)\) is a topological semigroup in spite of the fact that \(X\) is not locally compact.
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54D45
zbMATH DE Number: 3983509
zbMATH Keywords: semigroup of continuous selfmaps
zbMATH Keywords: compact-open topology
zbMATH Keywords: topological semigroup
zbMATH Keywords: locally compact
MaRDI profile type: Publication
cites work: A topology for spaces of transformations
cites work: Groups represented by homeomorphism groups. I
cites work: On topologies for function spaces
description: scientific article; zbMATH DE number 3983509