A countable free closed non-reflexive subgroup of $\mathbb {Z}^{\mathfrak {c}}$

Abstract:

We prove that the group $G=\mathrm {Hom}(\mathbb {Z}^{\mathbb {N}}, \mathbb {Z})$ of all homomorphisms from the Baer-Specker group $\mathbb {Z}^{\mathbb {N}}$ to the group $\mathbb {Z}$ 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 $G$ is discrete. As $G$ is non-discrete, it is not reflexive. Since $G$ can be viewed as a closed subgroup of the Tychonoff product $\mathbb {Z}^{\mathfrak {c}}$ of continuum many copies of the integers $\mathbb {Z}$, 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.