3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Bornological aspects of asymmetric structures(2021) Mukonda, DannyOver the last decades much progress has been made in the investigation of bornologies on metric spaces. In particular, Hu, Beer, Mero˜no, Garrido and others have published many papers on metric bornologies. The bornology of bounded sets in quasi-metric spaces was introduced by Pi¸ekosz and Wajch in 2015. They extended the Hu’s metrization theorem to quasi-metric spaces and applied it to bornologies of bitopological spaces. In 2019, Olela Otafudu et al. used the asymmetric version of Hu’s theorem to quasi-metrize the bornological universes on extended quasi-metric spaces. The principal aim of this thesis is to investigate the existence of bornologies of totally bounded sets and Bourbaki-bounded sets on asymmetric structures. In particular, we ex tend several results obtained by others on metric bornologies to quasi-metric settings. We show that a quasi-metric space can be bounded but not totally bounded and the bornology on a supseparable quasi-metric space agrees with the bornology of totally bounded sets. For Bourbaki-boundedness, it turns out that a set can be Bourbaki-bounded on a quasi-metric space but not on the metric space. In addition, we prove that every real-valued semi-Lipschitz in the small function is bounded if and only if the quasi-metric is Bourbaki bounded. Consequently, we use semi-Lipschitz functions to characterize those bornologies on asymmetric normed spaces that can be realized as bornolo gies of Bourbaki-bounded sets. For example, we show that on quasi-metric spaces, the bornology of Bourbaki-bounded sets sits between the bornology of totally bounded sets and the bornology of bounded sets but on asymmetric normed spaces, the bornology of Bourbaki-bounded sets coincides with the bornology of bounded sets.Item Some topological aspects of modular quasi-metric spaces(2019-06) Sebogodi, KatlegoIn this PhD thesis, we present the modular quasi-pseudometric on a nonempty set, a concept that generalises modular pseudometrics to the framework of quasi-metric spaces. We investigate the topological aspects of a set equipped with a modular quasi-pseudometric. It turns out that a set equipped with a modular quasi-pseudometric is a bitopological space in the sense of Kelly. Moreover, we introduce the theory of Isbell-convexity in the setting of modular quasi-pseudometrics which we shall call w-Isbell-convexity. Furthermore, we prove that given a modular set, w-Isbell-convexity is equivalent to Isbellconvexity whenever the modular quasi-pseudometric is continuous from the right on the set of positive numbers. We also study the boundedness of a set endowed with a modular quasi-pseudometric and we shall call it w- boundedness. Consequently, we show that a self w-nonexpansive map in a w-Isbell-convex modular set has a xed point and the set of xed points is w-Isbell-convex whenever the modular quasi-pseudometric is continuous from the right on the set of positive numbers.