Topological Study on Revised Fuzzy Metric Spaces and Their Generalization

Abstract

Introduction

In this paper, we explore the concept of metric functions within a revised fuzzy metric space. The study focuses on understanding the relationships between these metric functions and the topological structures they generate. Specifically, we introduce the concept of a stratified function within this framework and investigate its implications.

Objectives

The main objectives of this paper are:

1. To define and analyze stratified functions in a revised fuzzy metric space.


2. To demonstrate that the topology generated by the family of stratified functions coincides with the topology generated by the revised fuzzy metric.


3. To derive the concrete form of the metric function under specific conditions.

Methods

We approach these objectives by first introducing the notion of a stratified function in a revised fuzzy metric space. Using this concept, we prove that the topology generated by the family of stratified functions is identical to the topology generated by the revised fuzzy metric. Additionally, we explore the conditions under which a specific form of the metric function can be determined.

Results

Our findings show that the topology generated by the stratified functions indeed coincides with the topology generated by the revised fuzzy metric. Moreover, under certain special conditions, we can obtain a concrete representation of the metric function.

Conclusion

This paper provides a deeper understanding of the structure of revised fuzzy metric spaces. The introduction of stratified functions serves as a key tool for analyzing the topology of these spaces, and our results offer a concrete form for the metric function under specific conditions, contributing to the broader study of fuzzy metric spaces.