On Modular Quasi-Pseudometric Spaces

Abstract

In this MSc dissertation, we present modular metric spaces in asymmetric settings (called modular quasi-pseudometric spaces). Our results generalise and extend the concept of a modular metric on an arbitrary set, as presented in the work of Chistyakov. We show that Chistyakov's results also hold in an asymmetric framework. Furthermore, we show that most of Chistyakov's results do not need the symmetry property of a modular metric and prove that the results even hold when the symmetric property is not assumed. Finally, we observe that for any modular quasipseudometric which is convex on a set, its conjugate modular quasi-pseudometric is also convex and its symmetrized modular pseudometric preserves convexity.