Comparitive analysis of bornology in the categories of frolicher spaces and topological spaces
This thesis seeks to introduce the concept of bornology to the theory of Fr¨olicher spaces. Bornologies are induced from the Fr¨olicher structure, Fr¨olicher topology and the canonical topology on the underlying set of Fr¨olicher space. In each case the bornologies are compared in a general Fr¨olicher space, subspace, product, coproduct and quotient. The Fr¨olicher topology refers to the topology induced from structure functions of the Fr¨olicher space. The bornology induced from the Fr¨olicher structure is induced from the structure functions of Fr¨olicher space. An initial bornology is canonically induced on the underlying set of Fr¨olicher subspace and product, and a final bornology is induced canonically on the underlying set of Fr¨olicher coproduct and quotient. For Fr¨olicher subspace and product the initial bornology is finer than the bornology induced from structure functions. Dually, the final bornology is coarser than the bornology induced from structure functions for Fr¨olicher coproduct and quotient. Relatively-compact and compact bornologies are induced from the Fr¨olicher topology and the canonical topology on the underlying set of Fr¨olicher space. For each of Fr¨olicher subspace, product, coproduct and quotient, that is, the objects in the category of Fr¨olicher spaces under the study of this thesis, there are two topologies - the canonical topology induced from the underlying set and the Fr¨olicher topology. Subsequently there are two relatively-compact and compact bornologies, induced from these topologies, for each of the mentioned objects. The bornological comparison between the relatively-compact bornologies and the bornological comparison between the compact bornologies, for each object, is determined by the comparison of these topologies, that is, the comparison between the Fr¨olicher topology and the canonical topology on the underlying set.
A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2023
Bornology, Frolicher spaces, Topology