Some topological aspects of modular quasi-metric spaces

Abstract

In 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.