On a Generalization of Quasi-metric Space

TL;DR

This paper introduces the concept of g-quasi metrics, extending quasi-metrics to generalized topologies, and explores their properties, invariance, and extensions of classical metric concepts such as completeness and uniform continuity.

Contribution

It defines g-quasi metrics, investigates their topological invariance, and extends classical metric properties to this new framework.

Findings

g-quasi metrics can induce generalized topologies that may not be topologies

g-quasi metrizability is invariant under g-topological transformations

Classical metric properties like completeness are extended to g-quasi metric spaces

Abstract

We find an extension of the quasi-metric (to be called $g$-quasi metric) such that the induced generalized topology may fail to form a topology. We show that $g$-quasi metrizability is a $g$-topologically invariant property of generalized topological spaces. Extending metric product and uniform continuity for $g$-quasi metric spaces, we note that a $g$-quasi metric may fail to be uniformly continuous in the extended sense unlike usual metric. Finally, we extend the study of completeness, Lebesgue property and weak $G$-completeness for $g$-quasi metric spaces.