Let G be a maximally almost periodic (MAP) Abelian group and let B be a boundedness on G in the sense of Vilenkin. We study the relations between B and the Bohr topology of G for some well known groups with boundedness (G,B). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an L∞-group...