Bitopological Spaces Research Articles

Characterizations of generalized Fuzzy γ∗-closed sets

Generalized fuzzy open sets are playing a vital role in the study of fuzzy topological space as well as that of fuzzy bitopological space since its inception. More often, it is reported that fuzzy closed sets are always included in the family of generalized fuzzy closed sets. Very recently, it has appeared that fuzzy γ∗ -open sets are incomparable with fuzzy open sets. This paper aims to present three different kinds of fuzzy generalized closed sets in the light of fuzzy γ∗ -open set and associated closure operators with the terminologies- generalized fuzzy γ∗ -closed set, γ∗ -generalized fuzzy closed set and γ∗ -generalized fuzzy γ∗-closed set and it is found that the relation between any two concepts is not necessarily linear. Also, the interrelationships among them are established along with suitable counter examples which are properly placed to make the paper self-sufficient.

Some Subsets of Bitopological Spaces

In this paper, we introduce the notions of minimal u-ω-closed sets, maximal u-ω-open sets, u-ω-paraopen sets, u-ω-paraclosed sets, u-ω-mean open sets and u-ω-mean closed sets in bitopological spaces and obtain several characterizations and some of its properties.

Operation Pc-Open Sets and Operation Pc-Separation Axioms in Bitopological Spaces

Abstract In the present paper, we introduce new types of generalized closed sets called ij -pre-generalized closed sets and study some of their properties in bi-topological spaces. Also, we use them to construct new types of separation axioms. Further, we introduce and study the concepts of pairwise operation pc-open sets and pairwise operation pc-separation axioms in bitopological spaces. Several interesting characterizations of different spaces are discussed. The relationships between these spaces are given.

A Few Variants of Quasi-Continuity in Bitopological Spaces

Abstract The purpose of this paper is to introduce a few variants of generalized quasi-continuity of functions defined on a bitopological space and to study their mutual relationship. Moreover, some characterization of sectional quasi-continuous function and its continuity points are investigated.

Generalized Connectedness in Fuzzy Bitopological Spaces

Generalized Connectedness in Fuzzy Bitopological Spaces

Generalized Different Types of Mappings in Fuzzy Bitopological Spaces

The principle objective of this research is to present generalised functions ideas which are: fuzzy generalised continuous, generalized strongly continuous, generalized irresolute, generalized open and closed mapping, and the last part is the homomorphism in fuzzy bitopological spaces. We also study the relationships between them, their characteristics, their composition, and some important theories and counterexamples.

On c-Compactness in Topological and Bitopological Spaces

The primary goal of this research is to initiate the pairwise c-compact concept in topological and bitopological spaces. This would make us to define the concept of c-compact space with some of its generalization, and present some necessary notions such as the H-closed, the quasi compact and extremely disconnected compact spaces in topological and bitopological spaces. As a consequence, we derive numerous theoretical results that demonstrate the relations between c-separation axioms and the c-compact spaces.

Weakly Generalized β- Continuous Mapping in Neutrosophic Bitopological Spaces

This paper explores the notion of β-continuity in neutrosophic bitopological spaces, a specialized area of mathematics that extends classical topological concepts to handle indeterminate or uncertain information. The study begins with the introduction of τ₁τ₂ semi-closed sets and τ₁τ₂-weakly continuous functions, which are fundamental building blocks. Key results include Proposition 2.1.3, which characterizes τ₁τ₂-weakly β-continuous mappings in terms of pre-images and β-interiors of open sets in the codomain space. Propositions 2.1.4 and 2.1.5 establish equivalent conditions for τ₁τ₂-weakly β-continuous functions involving pre-images, closures, and regular closed sets. Propositions 2.1.6 and 2.1.7 provide alternative characterizations of τ₁τ₂-weakly β-continuous functions, revealing connections with β-interiors and pre-image relationships. These findings contribute to the understanding of topological properties in neutrosophic bitopological spaces, offering valuable insights for further research in this intricate field.

Pairwise β-Open Set in Neutrosophic Bitopological Spaces

This paper introduces the concepts of pairwise τ₁τ₂ neutrosophic-open sets, pairwise τ₁τ₂ neutrosophic-semi-open sets, and pairwise τ₁τ₂ neutrosophic-pre-open sets in neutrosophic bitopological spaces. We study some of the basic properties of these sets and prove several propositions, including the fact that the fusion of two τ₁τ₂ neutrosophic-open sets is a pairwise τ₁τ₂ neutrosophic-open set.

Note on Generalized Neighborhoods Structures in Fuzzy Bitopological Spaces

This article's main aim is to study the concepts of the generalized neighborhood and generalized quasi-neighborhood in fuzzy bitopological spaces. It also introduces fundamental theorems for determining the relationships between them. Additionally, some significant examples were examined to demonstrate the significance of the interconnections, some theorems were also introduced to study some main properties of neighborhood structures. Finally, we also studied the concepts of closure, interior, and each of their critical theories and properties by generalized neighborhood systems in fuzzy bitopological spaces.

Recognition of finite bitopological spaces with no isolated point

Recognition of finite bitopological spaces with no isolated point

Strongly separation axioms in fuzzifying bitopological spaces

Strongly separation axioms in fuzzifying bitopological spaces

Selection principles and bitopological hyperspaces

In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. For a Hausdorff space X we denote by 2X the family of all closed subsets of X. We investigate selection properties of the bitopological space (2X, Δ1+ , Δ2+) where Δi+ is the upper Δi-topology for each i=1,2.

Pairwise Semiregular Properties on Generalized Pairwise Lindelöf Spaces

Let (X, τ1, τ2) be a bitopological space and (X, τs(1,2), τs(2,1)) its pairwise semiregularization. Then a bitopological property P is called pairwise semiregular provided that (X, τ1, τ2) has the property P if and only if (X, τs(1,2), τs(2,1)) has the same property. In this work we study pairwise semiregular property of (i, j)-nearly Lindelöf, pairwise nearly Lindelöf, (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf spaces. We prove that (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf are pairwise semiregular properties, on the contrary of each type of pairwise Lindelöf space which are not pairwise semiregular properties.

Pairwise Neutrosophic Simply b-Open Set via Neutrosophic Bi-topological Spaces

In this article an attempt has been made to procure the concept of pairwise neutrosophic simply open set, pairwise neutrosophic simply continuous mapping, pairwise neutrosophic simply open mapping, pairwise neutrosophic simply compactness, pairwise neutrosophic simply b-open set, pairwise neutrosophic simply b-continuous mapping, pairwise neutrosophic simply b-open mapping and pairwise neutrosophic simply b-compactness via neutrosophic bi-topological spaces (in short NBTS). Besides, we furnish few illustrative examples on them via NBTS. Further, we investigate some basic properties of them, and formulate several results on NBTSs.

On r-Compactness in Topological and Bitopological Spaces

This paper defines the so-called pairwise r-compactness in topological and bitopological spaces. In particular, several inferred properties of the r-compact spaces and their connections with other topological and bitopological spaces are studied theoretically. As a result, several novel theorems of the r-compact space are generalized on the pairwise r-compact space. The results established in this research paper are new in the field of topology.

Soft i-Open Sets in Soft Bi-Topological Spaces

In our study we introduced soft i-open sets and soft i-star-generalized-w-closed sets in soft bi-topological spaces, , using the notion of soft i-open sets in soft-topological-space, . We besides that give examples to clarify these relationships while presenting some essential characteristics and relationships between various groups of sets. Besides that we studied the property of soft bi-topologically extended and non-soft bi-topologically extended of soft i-open sets in soft bi-topological spaces by proofs and examples.

On The Topolpgical Properties of Pairwise Compactness in Intuitionistic Double Topological Spaces

The concept of intuitionistic topological space was introduced by Coker. The aim of this paper is to discuss the relation between bi-topological spaces and double-topological spaces and give a notion of pairwise compact for double topological spaces.

A new class of NeutroOpen, NeutroClosed, AntiOpen and AntiClosed sets in NeutroTopological and AntiTopological spaces

A lot of research has been done on the types of open and closed sets in general topological spaces and also in general bitopological spaces. Types of sets like pre-open sets and pre-closed sets, semi-open sets and semi-closed sets, Alpha-open sets, and Alpha-closed sets, regular open sets and regular closed sets, g-open sets and g-closed sets, and many more have been defined and studied. In the current study, an attempt has been made to define and give examples of a new category of open and closed sets, namely, NeutroOpen and NeutroClosed sets. Further, the concept of neutron-topology is used to define NeutroPreOpen and NeutroPreClosed sets, NeutroSemiOpen and NeutroSemiClosed sets, NeutroAlphaOpen and NeutroAlphaClosed sets, NeutroRegularOpen and NeutroRegularClosed sets, NeutroBetaOpen, and NeutroBetaClosed sets, and several examples have been given to illustrate each of the new classes of sets. Also, the concept of AntiTopology has been used to define another class of sets, namely, AntiOpen and AntiClosed sets of the above five classes of sets, namely, regular-openclosed; semi-openclosed, Alpha-openclosed, Beta-openclosed pre-openclosed sets. Further, a new class of subsets is identified which are named as NeutroTauOpen and NeutroTauClosed sets. Similar subsets in anti-topological spaces are named as AntiTauOpen and AntiTauClosed sets.

Mappings and Finite Product of Pairwise Expandable Spaces

In this work, we intend to study certain mappings in bitopological spaces such as pairwise perfect and pairwise countably perfect mappings that possess the property (E). Some properties of such mappings are provided which have helped us to obtain some finite product theorems concerning with pairwise expandable and almost pairwise expandable spaces. Two illustrative examples are given to demonstrate the effectiveness of some proposed results.