Sebogodi, Katlego2020-09-152020-09-152019-06Sebogodi, Katlego, Some topological aspects of modular quasi-metric spaces, University of the Witwatersrand, Johannesburg, <http://hdl.handle.net/10539/29649>https://hdl.handle.net/10539/29649A thesis submitted to the Faculty of Science in fulfilment of the requirement for the degree Doctor of Philosophy (PhD) in Science, University of the Witwatersrand, Johannesburg, 2019In 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.Online resource (89 leaves)enFunction spacesMeasure theoryThesis