Bornological aspects of asymmetric structures

Abstract

Over 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.