Mukonda, Danny2021-12-132021-12-132021Mukonda, Danny (2021) Bornological aspects of asymmetric structures, University of the Witwatersrand, Johannesburg, <http://hdl.handle.net/10539/32299>https://hdl.handle.net/10539/32299A thesis submitted in fulfilment of the requirements for the degree of of Doctor of Philosophy (Mathematics) to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2021Over 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.Online resource (75 leaves)enFunction spacesMeasure theoryThesis