Note on reflexivity of some spaces of continuous integer-valued functions

Given a topological group G we denote by G ∧ the group of characters on G and reflexivity of G means that the natural map from G to G ∧ ∧ is a topological isomorphism. w that for any zero-dimensional realcompact k-space X and a discrete finitely generated abelian group A , the group A X of continuous maps from X to A with pointwise addition and compact–open topology is reflexive, and we construct a countable non-reflexive closed subgroup of Z X , where X is a countable subspace of the plane (this group embeds as a closed subgroup in many copies of the product ( ∑ Z ) c of continuum of the discrete Specker group). We show also that, for metrizable separable X , analyticity of ( Z X ) ∧ is equivalent to complete metrizability of X .