A countable free closed non-reflexive subgroup of ℤ^{𝔠}

Abstract

We prove that the group G = H o m ( Z N , Z ) G=\mathrm {Hom}(\mathbb {Z}^{\mathbb {N}}, \mathbb {Z}) of all homomorphisms from the Baer-Specker group Z N \mathbb {Z}^{\mathbb {N}} to the group Z \mathbb {Z} of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G G is discrete. As G G is non-discrete, it is not reflexive. Since G G can be viewed as a closed subgroup of the Tychonoff product Z c \mathbb {Z}^{\mathfrak {c}} of continuum many copies of the integers Z \mathbb {Z} , this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.