Bornological spaces

Bornology is a study of bounded sets, the spaces of such objects, and the naturally associated maps between the spaces of such objects. Previously, the setting for such a study has most often been in the presence of another type of structure such as metrics, topologies, uniformities, and/or, some sort of an algebraic structure with bounded sets being considered only as these were useful in the study of the main objects of concern. Here we carry out a study of bornology with its objects and maps and with these as the focus of the study. We describe the nature of the objects and maps of such things as subspaces, products, quotients, sums, etc., of the resulting category of those spaces. Mathematicians working on research projects centered around various kinds of topology related objects and maps will no doubt find the contents of the present paper very useful and efficient in that these mathematicians will not need to define boundedness related concepts or prove theorems about bounded sets since these items are within this present paper.