Note on reflexivity of some spaces of continuous integer-valued functions
type: Item
instance of: scholarly article
title: Note on reflexivity of some spaces of continuous integer-valued functions
zbMATH Open document ID: 1259.54006
author: R. Pol
author: Filip Smentek
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-08T00:00:00Z
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 6064299
zbMATH Keywords: reflexivity
zbMATH Keywords: abelian topological group
zbMATH Keywords: prodiscrete topological group
zbMATH Keywords: \(k\)-space
reviewed by: Vladimir Tkachuk
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.05.041
OpenAlex ID: W1979386642
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DOI: 10.1016/J.JMAA.2012.05.041
review text: Given topological spaces \(X\) and \(Y\), the symbol \(Y^X\) denotes the set of all continuous functions from \(X\) to \(Y\) endowed with the compact-open topology. If \(G\) is a topological group then \(G^X\) is also a topological group with the pointwise group operations. Let \(\mathbb T\) be the complex unit circle; then \(G^{\wedge}\) is the subgroup of \({\mathbb T}^G\) consisting of continuous homomorphisms, i.e., characters of \(G\). Let \(e(g)(\phi)=\phi(g)\) for any \(\phi\in G^{\wedge}\). Then \(e:G\to G^{\wedge\wedge}\) is called the evaluation map. A group \(G\) is called reflexive if its evaluation map is a topological isomorphism.NEWLINENEWLINEIt is proved in the paper that for any zero-dimensional realcompact space \(X\) and any discrete finitely generated abelian group \(A\), the group \(A^X\) is reflexive. Another result states that there exists a countable subspace \(X\) of the plane such that some countable closed subgroup of \({\mathbb Z}^X\) is not reflexive; here \(\mathbb Z\) is the group of integers. The authors also establish that for any separable metrizable space \(X\), the space \(({\mathbb Z}^X)^{\wedge}\) is analytic if and only if \(X\) is completely metrizable.
description: scientific article; zbMATH DE number 6064299