scholarly journals On T1 T2-g-Open Sets in Bitopological Spaces

2019 ◽  
pp. 1-8
Author(s):  
K. Vithyasangaran ◽  
P. Elango ◽  
S. Sathaananthan ◽  
J. Sriranganesan ◽  
P. Paramadevan
Keyword(s):  
Continuous Function ◽  
Topological Spaces ◽  
Bitopological Space ◽  
Open Set ◽  
Bitopological Spaces

In this paper, we introduced and studied a new kind of generalized open set called τ1τ2-g-open set in a bitopological space (X, τ1, τ2). The properties of this τ1τ2-g-open set are studied and compared with some of the corresponding generalized open sets in general topological spaces and bitopological spaces. We also dened the τ1τ2-g-continuous function and studied some its properties.

On pairwise \(\mathcal{N}\)-\(\beta\)-open sets in bitopological spaces

2016 ◽  
Vol 7 (3) ◽  
pp. 152
Author(s):  
Fahad Alsharari ◽  
Abdo Qahis
Keyword(s):  
Continuous Function ◽  
Open Cover ◽  
Bitopological Space ◽  
Open Set ◽  
Bitopological Spaces

In this paper, we introduce the notion of an \((i,j)\)-\(\mathcal{N}\)-\(\beta\)-open set which is a generalization of an \((i,j)\)-\(\beta\)-open set in a bitopological space. Also, we investigate some of its properties and characterizations. Besides, we prove that a pairwise \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-open cover that has a finite (countable) subcover is equivalent to a pairwise \(\beta\)-compact (\(\beta\)-Lindel\"{ö}f) space. Finally, we introduce an \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-continuous function and an \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-irresolute function and obtain some of their properties.

scholarly journals Neutrosophic Soft Bitopological Spaces

2021 ◽  
Author(s):  
Ahmed B. AL-Nafee ◽  
◽  
Said Broumi ◽  
Florentin Smarandache ◽  
◽  
...  
Keyword(s):  
Soft Set ◽  
Topological Spaces ◽  
Bitopological Space ◽  
Closed Set ◽  
Open Set ◽  
Bitopological Spaces

In this paper, we built bitopological space on the concept of neutrosophic soft set, we defined the basic topological concepts of this spaces which are N3-(bi)*-open set, N3-(bi)*-closed set, (bi)*-neutrosophic soft interior, (bi)* neutrosophic soft closure, (bi)*-neutrosophic soft boundary, (bi)*-neutrosophic soft exterior and we introduced their properties. In addition, we investigated the relations of these basic topological concepts with their counterparts in neutrosophic soft topological spaces and we introduced many examples.

New forms of separation spaces in bitopology

2016 ◽  
Vol 7 (3) ◽  
pp. 145
Author(s):  
N. Durga Devia ◽  
Raja Rajeswari ◽  
P. Thangavelu
Keyword(s):  
Continuous Function ◽  
Bitopological Space ◽  
Closed Set ◽  
Open Set ◽  
New Type

The aim of this paper is to study how distinct points and a point and a closed set not containing that points are separated by non overlapping open neighborhoods, in a bitopological space. The separation is studied with respect to a new type of \((1,2)\alpha\)-open set together with a continuous function. We named the new axioms as star-ultra \(T_{1}\), star-ultra \(T_{2}\), star-ultra regular and normal. The star-ultra regular spaces is studied in two different ways and are called as A-star-ultra regular and B-star-ultra regular spaces.

scholarly journals Projective bitopological spaces

1972 ◽  
Vol 13 (3) ◽  
pp. 327-334 ◽  
Author(s):  
M. C. Datta
Keyword(s):  
Projective Spaces ◽  
Topological Spaces ◽  
Bitopological Space ◽  
Extremally Disconnected ◽  
Bitopological Spaces

J. C. Kelly [2] introduced the concept of a bitopological space. Lane [3], Patty [4] and Pervin [5] have continued his work. Our purpose in this paper is to identify the projective objects in a suitable category of bitopological spaces after the manner of Gleason [1] and generalize his theorem that in the category of compact Hausdoriff topological spaces, the projective spaces are precisely the extremally disconnected ones.

scholarly journals Some Results of τ1τ2-δ Semiconnectedness and Compactness in Bitopological Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
M. Arunmaran ◽  
K. Kannan
Keyword(s):  
Topological Spaces ◽  
Bitopological Space ◽  
Bitopological Spaces ◽  
The Relationship

We are going to establish some results of τ1τ2-δ semiconnectedness and compactness in a bitopological space. Besides, we will investigate several results in τ1τ2-δ semiconnectedness for subsets in bitopological spaces. In particular, we will discuss the relationship related to semiconnectedness between the topological spaces and bitopological space. That is, if a bitopological space (X,τ1,τ2) is τ1τ2-δ semiconnected, then the topological spaces (X,τ1) and (X,τ2) are δ-semiconnected. In addition, we introduce the result which states that a bitopological space (X,τ1,τ2) is τ1τ2-δ semiconnected if and only if X and ϕ are the only subsets of X which are τ1τ2-δ semiclopen sets. Moreover, we have proved some results in compactness also. Altogether, several results of τ1τ2-δ semiconnectedness and compactness in a bitopological space have been discussed.

R1, Pairwise Compact, and Pairwise Complete Spaces

1972 ◽  
Vol 15 (1) ◽  
pp. 109-113 ◽  
Author(s):  
G. D. Richardson
Keyword(s):  
Topological Spaces ◽  
Bitopological Space ◽  
Closed Set ◽  
Usual Definition ◽  
Local Compactness ◽  
Definition Of ◽  
Bitopological Spaces

The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.

scholarly journals Characterization of Soft S-Open Sets in Bi-Soft Topological Structure Concerning Crisp Points

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2100
Author(s):  
Arif Mehmood ◽  
Mohammed M. Al-Shomrani ◽  
Muhammad Asad Zaighum ◽  
Saleem Abdullah
Keyword(s):  
Coordinate Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Finite Intersection ◽  
Open Set ◽  
Separation Axioms ◽  
Finite Intersection Property ◽  
Almost All ◽  
Bitopological Spaces

In this article, a soft s-open set in soft bitopological structures is introduced. With the help of this newly defined soft s-open set, soft separation axioms are regenerated in soft bitopological structures with respect to crisp points. Soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable and soft second-countable, soft connected, soft disconnected and soft locally connected spaces are defined with respect to crisp points under s-open sets in soft bitopological spaces. The product of two soft  axioms with respect crisp points with almost all possibilities in soft bitopological spaces relative to semiopen sets are introduced. In addition to this, soft (countability, base, subbase, finite intersection property, continuity) are addressed with respect to semiopen sets in soft bitopological spaces. Product of soft first and second coordinate spaces are addressed with respect to semiopen sets in soft bitopological spaces. The characterization of soft separation axioms with soft connectedness is addressed with respect to semiopen sets in soft bitopological spaces. In addition to this, the product of two soft topological spaces is (  space if each coordinate space is soft  space, product of two sot topological spaces is (S regular and C regular) space if each coordinate space is (S regular and C regular), the product of two soft topological spaces is connected if each coordinate space is soft connected and the product of two soft topological spaces is (first-countable, second-countable) if each coordinate space is (first countable, second-countable).

scholarly journals Fuzzy Homogeneous Bitopological Spaces

2018 ◽  
Vol 8 (6) ◽  
pp. 4619
Author(s):  
Samer Al Ghour ◽  
Almothana Azaizeh
Keyword(s):  
Bitopological Space ◽  
Open Set ◽  
Bitopological Spaces ◽  
Fuzzy Framework

We continue the study of the concepts of minimality and homogeneity in the fuzzy context. Concretely, we introduce two new notions of minimality in fuzzy bitopological spaces which are called minimal fuzzy open set and pairwise minimal fuzzy open set. Several relationships between such notions and a known one are given. Also, we provide results about the transformation of minimal, and pairwise minimal fuzzy open sets of a fuzzy bitopological space, via fuzzy continuous and fuzzy open mappings, and pairwise continuous and pairwise open mappings, respectively. Moreover, we present two new notions of homogeneity in the fuzzy framework. We introduce the notions of homogeneous and pairwise homogeneous fuzzy bitopological spaces. Several relationships between such notions and a known one are given. And, some connections between minimality and homogeneity are given. Finally, we show that cut bitopological spaces of a homogeneous (resp. pairwise homogeneous) fuzzy bitopological space are homogeneous (resp. pairwise homogeneous) but not conversely.

scholarly journals DISCUSSION ON QUOTIENT BI-SPACE AND ON PAIRWISE REGULAR AND NORMAL SPACES IN BITOPOLOGICAL SPACES

2019 ◽  
pp. 13-15
Author(s):  
M. Arunmaran ◽  
K. Kannan
Keyword(s):  
Compact Subset ◽  
Hausdorff Space ◽  
Bitopological Space ◽  
Closed Set ◽  
Continuous Map ◽  
Open Set ◽  
Bitopological Spaces

In this paper, we introduce the concept “Quotient bi-space” in bitopological spaces. In addition, we investigate the results related with quotient bi-space. Moreover, we have discussed the results related with pairwise regular and normal spaces in bitopological space. For a non-empty set X, we can define two topologies (these may be same or distinct topologies) τ1 and τ2 on X. Then, the triple (X, τ1 , τ2 ) is known as bitopological space. Let (X, τ1 , τ2 ) be bitopological space, (Y, σ1 , σ2 ) be trivial bitopological space and f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) be onto map. Then f is τ1 τ2 −continuous map. If η = {G (σ − open set in Y ) : f ^{−1} (G) is τ1 τ2 − open in X} then η is a topology on Y . Moreover, if (Y, σ, σ) be a quotient bi-space of (X, τ1 , τ2) under f : (X, τ1 , τ2 ) → (Y, σ, σ) and g : (Y, σ, σ) → (Z, η1 , η2 ) be a map, then, gis σ − continuous if and only if g ◦ f : (X, τ1 , τ2 ) → (Z, η1 , η2 ) is τ1 τ2 −continuous. Let (X, τ1 , τ2) be bitopological space and A be τ1 τ2 − compact subset of pairwise Hausdorff space X. Then, A is τ1 τ2 − closed set. Finally, we have discussed the following : Let (X, τ1 , τ2 ) be bitopological space and τ1 τ2 −compact pairwise Hausdorff space. Then, the space (X, τ1 , τ2 ) is pairwise normal.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

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