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.We show that for any zero-dimensional realcompact k-space X and a discrete finitely generated abelian group A, the group AX 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 ZX, 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 (ZX)∧ is equivalent to complete metrizability of X.