A boundedness structure (bornology) on a topological space is an ideal of subsets (that is, closed under taking subsets and unions of finitely many elements) containing all singletons, In this paper, we introduce certain functions (boundedness maps) as a tool in order to deal with the global pro\-per\-ties of families of bounded subsets rather than the specific properties of its members. Our motivation is twofold: on the one hand, we obtain useful information about the structural features of certain remarkable classes of boundedness systems, cofinality, local properties, etc. For example, we estimate the cofinality of these boundedness notions. In the second part of the paper, we apply duality methods in order to estimate the size of a local base for important classes of groups. This translation, which is well-known in the Pontryagin-van Kampen duality theory of locally compact abelian groups, is often very useful and has been extended by many authors to more general classes of topological groups. In this work we follow basically the pattern and terminology given by Vilenkin in 1998.