Bipolar Vague Contra ? Generalized Closed Mappings in Topological Spaces

Abstract

This paper is dedicated to the learning of bipolar vague topological spaces. In this paper we present the bipolar vague contra ? generalized closed mappings and bipolar vague contra ? generalized open mappings. Some of their belongings of bipolar vague contra ? generalized closed mappings and bipolar vague contra ? generalized open mappings are discussed.

Introduction

The paper explores bipolar vague sets and topologies, which extend fuzzy, intuitionistic fuzzy, and vague set theories by considering membership values in the range [-1, 1]. It begins by reviewing key historical developments in fuzzy, intuitionistic fuzzy, vague, and bipolar-valued fuzzy sets, and introduces bipolar vague topological spaces defined with positive and negative membership functions.

The authors define several types of bipolar vague sets (e.g., α-open, pre-open, semi-open, generalized closed/open sets) and their properties. They introduce bipolar vague contra α generalized closed mappings and bipolar vague contra α generalized open mappings, which are functions between bipolar vague topological spaces that preserve certain generalized openness or closedness properties under images of sets.

Examples illustrate these mappings, and propositions clarify their relationships—for instance, every bipolar vague contra closed mapping is also a bipolar vague contra α generalized closed mapping, but the converse does not always hold.

The paper contributes by presenting these new classes of mappings in bipolar vague topology and characterizing their basic properties, advancing the study of generalized topological structures in bipolar vague settings.

References

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Copyright

Copyright © 2025 F. Prishka, Dr. L. Mariapresenti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.