Browsing by Author "Mahudu, Ben Moditi"

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Comparitive analysis of bornology in the categories of frolicher spaces and topological spaces
(2024) Mahudu, Ben Moditi
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.
Topologies and smooth structures on initial and final objects in the category of frolicher spaces
(2019) Mahudu, Ben Moditi
The initial objects (in the category of Fro¨licher spaces) being studied are Fro¨licher subspace, product and equalizer’s domain; and the ﬁnal objects are Fro¨licher quotient, coproduct and coequalizer’s codomain. For each object a canonical topology (from the category of topologies) is induced on the underlying set of the object, and Fro¨licher topologies are induced from the Fr¨olicher structure. There are two Fr¨olicher topologies for each object: a Fro¨licher topology induced from structure curves and a Fr¨olicher topology induced from structure functions - it’s shown that the former Fr¨olicher topology is ﬁner than the latter Fr¨olicher topology for any Fr¨olicher space. It’s shown that for each initial object the canonical topology is coarser than the Fro¨licher topology induced from structure functions, and for each ﬁnal object the canonical topology is ﬁner than the Fr¨olicher topology induced from structure curves. Furthermore we establish that the building structure for each object is constant and algorithmic