An approach to duality on abelian precompact groups

Abstract

We prove that every dense Subgroup of a topological abelian group has the same 'convergence dual' as the whole group. By the 'convergence dual' we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence Structure on the character group Of a precompact group is discrete and therefore a non-compact precompact group Is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least M the category of metrizable abelian groups it does.