On the way to Frolicher bornology

Abstract

In this work we introduce the concept of bornology on the theory of Fr¨olicher spaces. A canonical bornology, called the functional  bornology, is induced from structure functions, and bounded maps under this bornology are determined. Based on the structure of  structure curves, we deduce that we cannot induce, canonically, a bornology from structure curves. For each of Fr¨olicher subspace and  product, an initial and functional bornology are induced and compared. Similarly, for each of Fr¨olicher quotient and coproduct, a final and  functional bornology are induced and compared.