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

TL;DR

This paper constructs a specific non-reflexive, countable, free, closed subgroup of Z^c, demonstrating that inverse limits of certain discrete abelian groups need not be reflexive, thus answering a previously posed problem.

Contribution

It provides an explicit example of a non-reflexive subgroup of Z^c, showing that inverse limits of finitely generated torsion-free abelian groups are not necessarily reflexive.

Findings

The group G has no infinite compact subsets.

The second Pontryagin dual of G is discrete.

G is a non-reflexive closed subgroup of a product of Z.

Abstract

We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group 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 of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby answering Problem 11 from [J.Galindo, L.Recorder-N\'{u}\~{n}ez, M.Tkachenko, Reflexivity of prodiscrete topological groups, J. Math. Anal. Appl. 384 (2011), 320--330.] It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.