Bitopological Spaces Research Articles

Semicontinuity, Semiconnected and Semicompactness in Bitopological Spaces

Let \(\tau_{1}\) and \(\tau_{2}\) be two topologies(same or distinct) which are defined in a nonempty set \(X\). Then, the triple \((X,\tau_{1},\tau_{2})\) is called as a bitopological space. The objective of this paper is to establish some results which are related with semi compactness in bitopological spaces and discuss the relationships between semi continuous function in bitopological space and various topological spaces. In particular, we identify the relationship between the bitopological spaces and their product space in semi compactness. Throughout this paper, we are able to get the clear understanding about the concept `semi compactness' and how to connect this concept with topological spaces and bitopological space. In addition, we can identify how to connect the continuous maps and product spaces with semi compactness. In addition, we can identify the relationships between semi continuous function in bitopological space and various topological spaces.

Pairwise Regularity and Normality Separation Axioms in C ̌ech Fuzzy Soft Bi-Closure Spaces

In this paper, some new types of regularity axioms, namely pairwise quasi-regular, pairwise semi-regular, pairwise pseudo regular and pairwise regular are defined and studied in both ech fuzzy soft bi-closure spaces ( bicsp’s) and their induced fuzzy soft bitopological spaces. We also study the relationships between them. We show that in all these types of axioms, the hereditary property is satisfied under closed fs bi-csubsp of . Furthermore, we define some normality axioms, namely pairwise semi-normal, pairwise pseudo normal, pairwise normal and pairwise completely normal in both bicsp’s and their induced fuzzy soft bitopological spaces, as well as their basic properties and the relationships between them are studied.

The assembly of a pointfree bispace and its two variations

We explore a pointfree theory of bitopological spaces (that is, sets equipped with two topologies). In particular, here we regard finitary biframes as duals of bitopological spaces. In particular for a finitary biframe \(\mathcal {L}\) the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly of a finitary biframe \(\mathcal {L}\), there are two other structures \(\mathsf {A}_{cf}(\mathcal {L})\) and \(\mathsf {A}_{\pm }(\mathcal {L})\), which both have the same main component as \(\mathsf {A}(\mathcal {L})\). The main component of both \(\mathsf {A}_{cf}(\mathcal {L})\) and \(\mathsf {A}_{\pm }(\mathcal {L})\) is the ordered collection of all biquotients of \(\mathcal {L}\). The structure \(\mathsf {A}_{cf}(\mathcal {L})\) being a biframe shows that the collection of all biquotients is generated by the frame of the patch-closed biquotients together with that of the patch-fitted ones. The structure \(\mathsf {A}_{\pm }(\mathcal {L})\) being a biframe shows the collection of all biquotients is generated by the frame of the positive biquotients together with that of the negative ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of two characterization theorems for these axioms hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise \(T_1\) are characterized, among other things in terms of the spectrum \(\mathsf {bpt}(\mathsf {A}_{cf}(\mathcal {L}))\).

Neutrosophic SuperHyper Bi-Topological Spaces: Original Notions and New Insights

Neutrosophic SuperHyper Bi-Topological Spaces: Original Notions and New Insights

Countably and Locally Compactness in Bitopological Spaces

The main objectives of this paper are to introduce the concept of pairwise countably compact in bitopological space,and we introduce some generalization of pairwise countably compact in bitopological spaces, and some of their properties, and relate it to other spaces.

ON ROUGH BI-SEMI GENERALIZED LOCALLY CLOSED SETS IN ROUGH SET BITOPOLOGICAL SPACES

The purpose of this paper is to introduce a new class of sets called Rough bi-semi locally closed sets and Rough bi-semi generalized locally closed sets in Rough bitopological space and studied some of its properties. Keywords: Rough bitopology, Rough bi-closed Sets, Rough bi-sg closed Sets, Rough bi-SLC Sets, Rough bi-SLC* Sets,, Rough bi-SGLC Sets, Rough bi-SGLC* set, Rough bi-SGLC** set

On Pairwise P-closed Spaces

In this paper, we introduce the concept of pairwise P-closed spaces and obtain some of their properties. Furthermore, we generalize some results concerning P-closed spaces to pairwise P-closed. Eventually, we conclude that every p-paracompact, p.w.T2 bitopological space is p-normal.

ℝ𝑶𝑼𝑮𝑯 𝓸𝑯(𝟏,𝟐)∗ GENERALIZED PRE-HOMEOMORPHISMIN ROUGH BITOPOLOGICAL SPACE

The aim of this paper is to introduced and study a new class of functions called ℝ𝑜𝑢𝑔ℎℴ𝐻(1,2)∗ generalized pre-homeomorphism function (briefly ℝℴ𝐻(1,2)∗𝑔𝑝ℎ𝑓 ) in Bitopological Space and to analysis some properties and its relation with existing functions are studied. Keywords: ℝ ℴ𝐻(1,2)∗𝑔𝑝ℎ𝑓 , ℝℴ𝐻(1,2)∗gp – C, ℝℴ𝐻(1,2)∗ 𝑔𝑝 – closed set, ℝℴ𝐻 (1,2)∗- open set

Regularity for bitopological spaces

Regularity for bitopological spaces

On Pairwise C-closed Topological Spaces

Ismail first proposed the concept of C-closed topological spaces in 1980, assuming that spaces which are countably compact are closed. In 2019, Omar and Hdeib introduced the notion of pairwise C-closed bitopological spaces. In this article, several results concerning these notions are proposed and discussed. Many findings are summarized in relating pairwise strongly Lindelöf bitopological space and pairwise strongly C-lindelöf.

Generalized $\delta$-$s\bigwedge _{ij}$-sets in bitopological spaces

The concepts of $ij-\delta-$semi closed and $ij-\delta-$semi open sets in bitopological spaces are introduced and studied. Also, the notions of {$\delta -s\bigwedge _{ij} -$}sets and {$g\delta -s\bigwedge _{ij} -$}sets are investigated. Furthermore , a new closure operator called $Cl_{\delta }^{s\bigwedge _{ij} } $ on the bitopological space $(X,\tau _{1},\tau _{2})$ is defined and associated topology $\tau _{\delta }^{s\bigwedge _{ij} }$ is given.

Neutrosophic soft bi-topological structures concerning some special neutrosophic soft open sets

In this article, new generliased neutrosophic soft *b open set is introduced in neutrosophic soft bi-topological spaces (NSBTS) concerning soft points of the space. This new set is produced by the combination of soft semi-open and soft pre-open sets of neutrosophic soft topological space. Different results are ushered in NSBTS. Appropriate examples are provided for verification of different results. The non-validity of some results is verified with appropriate examples.

Neutrosophic pre-I-open set in neutrosophic ideal bitopological space

Neutrosophic pre-I-open set in neutrosophic ideal bitopological space

NEW CONCEPTS ON R1 FUZZY SOFT BITOPOLOGICAL SPACE IN QUASI-COINCIDENCE SENSE

In this paper, three notions of property in fuzzy soft bitopological spaces in the sense of quasi-coincidence for fuzzy soft points has been introduced and studied. Hereditary, productive, and projective properties are satisfied by these notions. Moreover, it is observed that all these concepts are preserved under one-one, onto, fuzzy open, and FSP continuous mappings.

Bitopological spaces in the context of pythagorean fuzzy set

Bitopological spaces in the context of pythagorean fuzzy set

Preopenness Degree in R L -Fuzzy Bitopological Spaces

Based on the concept of pseudocomplement, we introduce a new representation of preopenness of L -fuzzy sets in R L -fuzzy bitopological spaces. The concepts of pairwise R L -fuzzy precontinuous and pairwise R L -fuzzy preirresolute functions are extended and discussed based on the i , j - R L -preopen gradation. Further, we follow up with a study of pairwise R L -fuzzy precompactness in R L -fuzzy bitopological spaces of an L -fuzzy set. We find that our paper offers more general results since R L -fuzzy bitopology is a generalization of L -bitopology, R L -bitopology, and L -fuzzy topology.

Generalized Open Sets in Neutrosophic Soft Bitopological Spaces

Generalized Open Sets in Neutrosophic Soft Bitopological Spaces

The category of finitary biframes as the category of pointfree bispaces

The theory of finitary biframes as order-theoretical duals of bitopological spaces is explored. The category of finitary biframes is a coreflective subcategory of that of biframes. Some of the advantages of adopting finitary biframes as a pointfree notion of bispaces are studied. In particular, it is shown that for every finitary biframe there is a biframe which plays a role analogue to that of the assembly in the theory of frames: for every finitary biframe L there is a finitary biframe A ( L ) with a universal property analogous to that of the assembly of a frame; and such that its main component is isomorphic to the ordered collection of finitary quotients of L (i.e. its pointfree bispaces). Furthermore, in the finitary biframe duality the bispace associated with A ( L ) is a natural bitopological analogue of the Skula space of the bispace associated with L . The finitary biframe duality gives us a notion of bisobriety which is weaker than pairwise Hausdorffness, incomparable with the pairwise T 1 axiom, and stronger than the pairwise T 0 axiom. The notion of pairwise T D bispaces is introduced, as a natural point-set generalization of the classical T D axiom. It is shown that in the finitary biframe duality this axiom plays a role analogous to that of the classical T D axiom for the frame duality.

(1,2) *-D*-closed sets and (1,2) *-D**-closed sets in ideal bitopological spaces

In this paper, we introduce the notion of (1,2)*-D*-closed sets and (1,2)*-D**-closed sets in ideal bitopological spaces. Additionally, we examine the characteristics of the novel concepts and contrast them with preexisting ones.

On almost set-Menger spaces in bitopological context

<abstract><p>In this paper, we define the $ ij $-almost-set Menger ($ ij $-ASM) property in bitopological spaces. We put up some equivalences of $ ij $-almost-set Menger bitopological spaces and investigate the behaviours of such spaces under some different types of mappings. We later take the preservation of these properties under union, subspaces, products into consideration and give some related examples. We finally introduce the concept of $ ij $-almost $ P_{\gamma} $-set in bitopological spaces.</p></abstract>