An Analysis on Fuzzy E -Open sets and Fuzzy E -Continuity in Topological Spaces

Reeja P S ¹ * , Dr. Dharmendra Saxena ²

1. Research Scholar, University of Technology, Jaipur, Rajasthan, India
reejaps2012@gmail.com ,

2. Professor, Department of Mathematics, University of Technology, Jaipur, Rajasthan, India

Abstract: Fuzzy set theory extends traditional topological concepts through Fuzzy E-Open Sets and Fuzzy E-Continuity. These sets, in conjunction with fuzzy e-open sets, are generalised variations of conventional topological sets such as β-, a-, and e-open sets. Fuzzy and intuitionistic fuzzy e-open sets exhibit topological patterns, however their interrelations and behaviours may not consistently be reciprocal.

Keywords: Fuzzy E -Open Sets , Fuzzy E -Continuity , Topological Spaces

INTRODUCTION

Fuzzy set theory allows for membership degrees to range from 0 (non-member) to 1 (member), and it was originally introduced a few decades ago. This methodology is advantageous for simulating ambiguity and uncertainty across several fields, including computer science, engineering, and mathematics.

Fuzzy E-Openness and E-Continuity

1. Generalization: They are beneficial in contexts where exact categorisation or membership is unattainable, allowing items to possess different degrees of affiliation to a set or a continuous mapping..

2. Thresholding: The primary distinction from classical topology is the incorporation of the threshold ε, which enables a fuzzy definition of "closeness" or "openness," as opposed to the rigid classification of sets as either open or closed..

3. Applications: These notions are applicable in contexts including ::

Artificial Intelligence and Machine Learning: Modelling uncertainty or ambiguity in decision-making processes.

o Fuzzy Control Systems: Systems that function amidst ambiguity and necessitate adaptable decision thresholds.

o Image Processing: In instances when pixel values do not distinctly belong to a single category (e.g., object vs backdrop), fuzzy sets can effectively express partial memberships.

OBJECTIVES OF THE STUDY

1. To study on Fuzzy topological space

2. To study on Fuzzy e-continuity and separation axioms for open sets

METHODOLOGY

We examined the methodology for fuzzy E-continuity in topological spaces and fuzzy E-open sets. The analysis of fuzzy E-continuity was also performed using severance axioms.

DATA ANALYSIS

CONCLUSION

An open set in classical topology is a subset of a topology, which is a collection of sets that satisfies particular axioms. Fuzzy sets by using fuzzy membership values. Fuzzy e-continuous mappings are presented and studied, focussing on separation axiom characteristics..