Closed ideals in semigroups of continuous selfmaps
type: Item
instance of: scholarly article
title: Closed ideals in semigroups of continuous selfmaps
zbMATH Open document ID: 0746.54016
DOI: 10.1007/BF02574341
author: A. M. Forouzanfar
published in: Semigroup Forum
publication date: 1992-06-28T00:00:00Z
full work available at URL: https://eudml.org/doc/135145
review text: All topological spaces are assumed to be Hausdorff and \(S(X)\) is the semigroup of all continuous selfmaps of the topological space \(X\). The author provides \(S(X)\) with the topology of pointwise convergence. \(S(X)\) is not, in general, a topological semigroup with this topology. Indeed, if \(X\) is locally compact, then the compact-open topology is the smallest topology for which \(S(X)\) is a topological semigroup and the topology of pointwise convergence is generally smaller than the compact-open topology. Nevertheless, the topology of pointwise convergence still is, as the author points out, a natural topology for \(S(X)\) in that one does have a semitopological semigroup and this will be the topology on \(S(X)\) for the remainder of the discussion. The author shows that for certain spaces \(X\), the closure of the principal right ideal generated by \(f\in S(X)\) is \(\{g\in S(X): g[X]\subseteq\text{cl} f[X]\}\) and he shows that all principal left ideals are closed. He then goes on to describe those left and right ideals of \(S(I)\) (\(I\) is the closed unit interval) which are closed and he uses the previous results to show that the only closed two-sided ideals of \(S(X)\) are the ideal of all constant functions and \(S(X)\) itself. And now, some further comments are in order. First of all, Proposition 2.6 is false. It asserts that if \(X\) is completely regular and locally pathwise connected, then each map from a finite subset of \(X\) into \(X\) can be extended to a continuous selfmap of \(X\). In fact, it is not difficult to verify that if \(X\) is completely regular and contains an arc, then each map from a finite subset of \(X\) into \(X\) can be extended to a continuous selfmap of \(X\) if and only if \(X\) is pathwise connected. The error, however, is not at all a serious one. It just means that the class of spaces to which the other results apply is somewhat less extensive. As for the principal right ideals of \(S(X)\), one can show for any space …
reviewed by: jun. Kenneth D. Magill
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 35817
zbMATH Keywords: principal left ideals
zbMATH Keywords: closed two-sided ideals
zbMATH Keywords: principal right ideals
zbMATH Keywords: regular element
MaRDI profile type: Publication
cites work: The semigroup of continuous selfmaps ofIhas infinitely many ideals
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cites work: On a family of ideals of S(X)
cites work: Topologies on Spaces of Subsets
OpenAlex ID: W2094204352
description: scientific article; zbMATH DE number 35817