Note on reflexivity of some spaces of continuous integer-valued functions
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.
scientific article; zbMATH DE number 6064299
scientific article