# Introducing a new concept of distance on a topological space by generalizing the definition of quasi-pseudo-metric

## TL;DR

This paper introduces a new generalized R.O-metric structure on topological spaces, extending quasi-pseudo-metrics, and proves that all topological spaces can be endowed with this structure.

## Contribution

It defines the generalized R.O-metric space, broadening the concept of distance in topology, and shows that every topological space is compatible with this new structure.

## Key findings

- Definition of R.O-metric space and its properties
- Introduction of generalized R.O-metric space
- Proof that all topological spaces are generalized R.O-metrizable

## Abstract

In this paper, a new structure is defined on a topological space that equips the space with a concept of distance in order to do that firstly, a generalization of quasi-pseudo-metric space named R.O-metric space is introduced, and some of its basic properties is studied. Afterwards the concept of generalized R.O-metric space is defined .Finally, we establish that every topological space is generalized R.O-metrizable.