A characterization of the MAP abelian groups

We introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group.We prove that a topological abelian group G is maximally almost periodicif and only if every cyclicsubgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97–113, A characterization of the circle group and the p-adicinte gers via sequential limit laws, preprint), and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adicinte gers via sequential limit laws, preprint).