Bitopological Spaces Research Articles

A new Outlook on Omega Closed Functions in Bitopological Spaces and Related Aspects

Many studies have employed a variety of techniques to further investigate topological space, particularly the notion of bitopological spaces, due to the significance of topological space in data processing as well as certain implementations. Numerous extended topological structures have been laid out subsequently. Of those abstractions, functions in topology was one of which was most noteworthy. In order to assist in this trend, we focused our research on the idea of open and closed sets, which is one of the strongest techniques available to present scientists for the study of computer graphics and digital topology.New functions, pairwise ω−closed functions, which are strictly weaker than pairwise closed functions, will be introduced in this study. By applying the P˝−space definition, whose is a P−space modification. Additionally, we establish different projection and product theories pertaining to pairwise Lindelöf of and pairwise paracompact spaces utilizing P˝−spaces. We analyze images and inverse images that have been chosen topological attributes for every one of these functions. In the final analysis, we explore several counterexamples that correspond to the offered definitions and theorems.

Fuzzy Topological Spaces Generated by Fuzzy Digraphs

In this paper, we defined various fuzzy topological spaces on fuzzy digraph and study interrelation between them. Using adjacency relation on fuzzy vertices and fuzzy edges of fuzzy digraph we defined, namely two types of fuzzy topologies, left(right) fuzzy vertex topology and left(right) fuzzy edge topology on fuzzy digraph, respectively. We have obtained results related to fuzzy topology on fuzzy subdigraph [Formula: see text] of fuzzy digraph [Formula: see text]. Separation axioms [Formula: see text], [Formula: see text] and [Formula: see text] for these fuzzy topological spaces are studied. Some characterization results for fuzzy topological spaces related to isomorphic fuzzy digraphs are given. Further, we have shown that two fuzzy digraphs are isomorphic if and only if their corresponding fuzzy topologies are homeomorphic, this relation on set of fuzzy digraphs forms an equivalence relation. We have also defined and studied fuzzy bitopological spaces on fuzzy digraphs.

Operation on Generalized Semi Open Sets in Bitopological Spaces

The aim of this paper is to more contribute the study of operations on bitopological spaces. The concept of operation on generalized semi open sets in bitopological spaces is introduced and studied. Two closure operators related to this concept are introduced and some of their properties and the relation between them are discussed.

SOME FEATURES OF PAIRWISE α-R0 SPACES IN SUPRA FUZZY BITOPOLOGY

This paper introduces and studies four concepts of supra fuzzy bitopological spaces. We have exhibited that all these four concepts are ‘good extensions’ of the corresponding concepts bitopological spaces and building relationships among them. It has been justified that all the definitions are hereditary, productive, and projective. Furthermore, additional properties of these concepts are studied.

Intuitionistic fuzzy normal bi-topological space-approach of intuitionistic fuzzy open sets

This paper provides nine newly proposed notions of intuitionistic fuzzy normal bi-topological spaces (IFNBTS) based on the concept of most explored field fuzzy bi-topological spaces using intuitionistic fuzzy open sets (IFOS). Further, the authors establish implications among the prescribed notions and show that these notions are good extensions of normal and fuzzy normal bi-topological spaces. Finally, the authors study the image and pre-image of IFNBTS, demonstrating that they are also IFNBTS in the sense of IFOS.

Strongly 𝑫𝜶𝒑 −closed graphs in bitopological spaces

In this paper, we introduce the concepts of D_α^p-open, D_α^p-closed subsets, pairwise-α-closed, pairwise-g-closed subsets, pairwise-strongly α-closed graph G(f) and strongly D_α^p-closed graph of bitopological spaces. We showed that each closed graph is D_α^p-closed. In addition, the concepts of D_α^p-continuous, open, and closed functions are defined, and the relations between τ_p-α, τ_p-g, and D_α^p-continuous functions are clarified. The fact that strongly D_α^p-closed graph is D_α^p-closed is illustrated. We studied when the graph G(f) is p-strongly closed and p-D_α-closed subsets of the bitopological space (X,τ₁,τ₂). Moreover, the notions of D_α^i-interior of a subset of X and D_α^i-closure are defined.

Separation axioms on fuzzy neutrosophic bitopological spaces using fuzzy neutrosophic open sets

AbstractIndeterminacy, non-membership and membership, have been studied in a neutrosophic set (NS). Using the notion of NS, neutrosophic topological space (NTS) and neutrosophic bitopological spaces have been developed. The well-known properties (separation axioms) of neutrosophic $${\text{T}}_{i}$$ T i -topological and bitopological spaces $$(i=0, 1, 2)$$ ( i = 0 , 1 , 2 ) are different types of neutrosophic spaces with different characteristics. Here, we define the idea of fuzzy neutrosophic $${\text{T}}_{i}$$ T i -bitopological space $$(i=0, 1, 2)$$ ( i = 0 , 1 , 2 ) through the neutrosophic bitopological space and look into its various characteristics. From neutrosophic $${\text{T}}_{i}$$ T i -bitopological space ($$i=0, 1, 2$$ i = 0 , 1 , 2 ), we demonstrate some intriguing findings with examples about the neutrosophic separation axioms.

On $\psi$ gs-Functions in Bitopological Spaces

A subset $A$ of a bitopological space $(X,\tau_1,\tau_2)$ is called \emph{$(i,j)$-$\psi$gs-closed} set if\\ $(i,j)\text{-}\psi cl(A)\subseteq U$ whenever $A\subseteq U$, $U$ is $(i,j)$-semi-open in $(X,\tau_1,\tau_2)$. In this work, the propertiesof this set are considered to investigate the concepts of $\psi gs$-functions in bitopological spaces. Specifically, this study establishes some properties and provide characterizations of $\psi gs$-open and $\psi gs$-closed functions, $\psi gs$-continuous functions, and $\psi gs$-irresolute functions in bitopological spaces.

Application of operation approach in fuzzy bitopological spaces

Application of operation approach in fuzzy bitopological spaces

An in-depth exploration of intuitionistic fuzzy T_0 in the context of bitopology

Intuitionistic fuzzy topological space and bitopological space have been introduced by using the concepts of intuitionistic fuzzy sets which are the generalizations of interval valued fuzzy sets. This paper commences by presenting the notion of intuitionistic fuzzy T0 in the context of bitopological spaces (IFB-T0). Subsequently, we explore various connections and relationships between these concepts. Then, we find out the relation between intuitionistic T0 and IFB-T0 spaces. Further, we investigated continuity between two IFB spaces.

Some subsets in bitopological spaces and bioperations continuity associated

The aim of this paper is to introduce and study some new types of sets via bioperations and to define and characterize some new forms of bioperations continuity using these sets and their relationships.

Lower separation axioms in bitopogenous spaces

Abstract Several naturally defined lower separation axioms for bitopological spaces obtained by modifying the axioms T 0, T 1, and R 0 appear in the literature. We introduce and study analogous separation axioms for bitopogenous spaces. In particular, we investigate relationships between the axioms and discuss the conditions under which the relationships are similar to those between the corresponding separation axioms for bitopological spaces.

Neutrosophic SuperHyper Bi-Topological Spaces: Extra Topics

This manuscript holds the integral concepts presented by the previously published article called (Neutrosophic SuperHyper Bi-Topological Spaces). This essay contains new definitions related to the notion of SuperHyper Bi-Topological Spaces, such as SuperHyper Supra closure operator, the pairwise neutrosophic interior of the power set Pn1(x), three cascade numerical examples demonstrate the NSHBI-TS, pairwise neutrosophic nth-power sets in (τ11ndpair, τ22ndpair,Pn(x)), as well as extra definitions, theorems, corollaries, and propositions.

On Generalized Compactness in Fuzzy Bitopological Spaces

The main objective of this research is to study some types of generalized closed sets in fuzzy bitopology including $(i,j)-g\alpha-cld$, $(i,j)-gs-cld$, $(i,j)-gp-cld$, and $(i,j)-g\beta-cld$. We then present basic theorems for determining their relationships and explain their properties, such as closure and interior. In addition, there are many interesting counterexamples. The last part of the research focuses on generalized compactness in fuzzy bitopological spaces and their types and explores the relationships between these concepts, their important theories, and some relevant counterexamples. The results established in this paper are new in the domain of fuzzy bitopology.

$ KC $-bitopological spaces

<p>A topological space $ \left(X, \tau \right) $ is called a $ KC $-space when every compact subset of $ X $ is closed. The aim of this paper is to introduce new, namely $ KC $-bitopological spaces and pairwise $ KC $-topological spaces "$ P $-$ KC $-topological spaces". We examined the properties of these concepts and showed the relationships between these concepts and other bitopological spaces. We also discussed the effect of some types of functions on $ KC $-bitopological spaces and pairwise $ KC $-topological spaces. Several examples are discussed, and many well-known theories are generalized.</p>

On Some Separation Axioms in Bitopological Spaces

On Some Separation Axioms in Bitopological Spaces

Some Properties of Fuzzy Semi-Open Sets in Fuzzy Bi-Spaces

The idea of fuzzy “semi-open sets” within the framework of fuzzy fields was proposed in the theory of fuzzy topology. This investigation delves deeper into the concept, specifically examining fuzzy “semi-open sets” concerning ω̃1 (ω̃2) with respect to ω̃2 (ω̃1). Additionally, we explore pairs of fuzzy “semi-open sets” in the context of a fuzzy bi-space and analyse their implications on results applicable in bi-topological spaces.

Single-Valued Pentapartitioned Neutrosophic Bi-Topological Spaces

In this article, we present the notion of Single-Valued Pentapartitioned Neutrosophic Bi-Topological Space (SVPNBTS) as a generalization of Single-Valued Pentapartitioned Neutrosophic Topological Space (SVPNTS) and Neutrosophic Bi-Topological Space (NBTS). Besides, we study the different types of open set and closed set namely single-valued pentapartitioned neutrosophic bi-open set (SVPNBOS), single-valued pentapartitioned neutrosophic bi-closed set (SVPNBCS), single-valued pentapartitioned neutrosophic bi-semi-open set (SVPNBSOS), single-valued pentapartitioned neutrosophic bi-semi-closed set (SVPNBSCS), single-valued pentapartitioned neutrosophic bi-pre-open set (SVPNBPOS), single-valued pentapartitioned neutrosophic bi-pre-closed set (SVPNBPCS), single-valued pentapartitioned neutrosophic bi-b-open set (SVPNBb-OS), single-valued pentapartitioned neutrosophic bi-b-closed set (SVPNBb-CS), etc. via SVPNBTSs. Besides, we introduce the notion of pairwise SVPNOS, pairwise SVPNCS, pairwise SVPNSOS, pairwise SVPNSCS, pairwise SVPNPOS, pairwise SVPNPCS, pairwise SVPNb-OS, pairwise SVPNb-CS, and furnish few illustrative examples on them. Further, we investigate several properties of these classes of sets and prove some interesting results in the form of propositions, theorems, etc. via SVPNBTSs.

Advanced Decision-Making Techniques with Generalized Close Sets in Neutrosophic Soft Bitopological Spaces

In recent years, neutrosophic soft bitopological spaces have emerged as a promising framework for handling uncertainty and imprecision in various domains, particularly in the context of decision-making problems. This paper presents a comprehensive study of advanced decision-making techniques using generalized close sets in neutrosophic soft bitopological spaces. The primary objective of this research is to develop a better understanding of the theoretical underpinnings of generalized close sets and their practical applications in decision-making under uncertain conditions. We begin by providing a detailed introduction to the key concepts of neutrosophic sets, neutrosophic soft sets, and neutrosophic soft bitopological spaces. Subsequently, we introduce generalized close sets and discuss their properties and interrelationships with other relevant constructs in the field. The paper then delves into the decision-making aspect, presenting various methodologies for solving decision-making problems using generalized close sets in the context of neutrosophic soft bitopological spaces. Numerous illustrative examples and case studies are provided throughout the paper to demonstrate the applicability and effectiveness of the proposed techniques in handling complex decision-making problems in real-world scenarios. The results of this research not only contribute to the existing body of knowledge in the field but also offer valuable insights for practitioners and researchers seeking to employ advanced decision-making techniques in uncertain environments.

Normal Separation Axiom on Fuzzy Bitopological Space

In our paper, we present three novel concepts related to the normal separation property within the realm of fuzzy bitopological spaces (FPTS), specifically focusing on pairwise fuzzy normal bitopological spaces (FPN). These notions are introduced in a quasi-coincidence sense, and we establish relationships between our propositions and other existing ones. Furthermore, we provide proofs demonstrating that all the introduced concepts exhibit the ‘good extension’ property. Notably, we observe that our notions maintain their characteristics under one-one, onto, fuzzy pairwise open (FP-open), fuzzy closed (FP-closed), and fuzzy pairwise continuous (FP-continuous) mappings.