On a generalization of quasi-metric space

We introduce a distinctive metric structure for generalized topology by extending the quasi-metric. This extension (to be called gquasi metric) naturally induces a generalized topology, yet it may diverge from forming a topology. We demonstrate that g-quasi metrizability remains an invariant property of generalized topological spaces. Expanding the concepts of metric product and uniform continuity within g-quasi metric spaces, we observe an instance where a g-quasi metric may not exhibit uniform continuity like standard metrics. Additionally, we study completeness, Lebesgue property, and weak G-completeness for g-quasi metric spaces.