Separation Axioms Research Articles

Some Separation Axioms via Soft Somewhat Open Sets

It is commonly known that some topological spaces include structures that may be used to expand abstract notions. somewhat open sets and soft sets is such sort of structures. We obtain several properties and symmetry of the soft somewhat-R0 spaces and soft somewhat-R1 spaces obtained. Furthermore, we present new theorems and results and investigated relation between this concepts and the other structures.

Some New Types of Fuzzy Closed Sets, SeparationAxioms, and Compactness via Double Fuzzy Topologies

In this article, we first defined a stronger form of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

The Degree-Topology of CertainTypes of Graphs and PT0-Space

This work aimed to study the degree topology of certain types of graphs namely, the wheel, helm, gear, pan, tadpole, and windmill graphs, respectively. In addition, a new separation axiom known as -space that joins the topological space and paths in the graph is introduced. Also, the relation between -space and the degree-topology of the connected and disconnected graph was stydied. Moreover, many examples were studied in new separation space.

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.

On Intuitionistic Fuzzy Soft Pretopology

This paper defines the following terms: intuitionistic fuzzy soft (IFS) pretopological space, IFS interior function, IFS pre-open set, IFS pre-closed set, trace of a IFS pretopology, IFS separation axioms, IFS subspace, IFS compactness, IFS connectedness, and some of their properties. In addition, the degree of soft non-vacuity, the soft α-cut, and the IFS preneighbourhood system at a soft point are defined. IFS preneighbourhoods produce IFS pretopologies.

On traditional Menger and Rothberger variations

We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to w-covers and k-covers in the most general topological setting and address the finite productivity of some of these properties. We collect various examples that separate certain properties and we carefully identify which separation axioms simplify aspects of these properties. We finish with a consolidated list of open questions focused on topological examples.

Lower spaces of multiplicative lattices

AbstractWe consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, including lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space; we further explore continuous maps between lower spaces.

K- Hausdorff Separation Axiom in Second Order Interval Valued Fuzzy Topological Spaces

We studied and introduced a concept SIVFT then present the concept of SIVF subspace and SIVF product topology in SIVF topological spaces. K-Hausdorff Separation Axiom in SIVF topological spaces and its basics are studied.

Correspondence of Fixed-Point Theorem in \(T_2, T_3\)-SPACE

Fixed-point theory (FPT) has lot of applications not only in the field of mathematics but also in various other disciplines. Fixed Point Theorem presents that if T:X \(\to\) X is a contraction mapping on a complete metric space (X, d) then there exists a unique fixed point in X. FPT is also essential in game theory, in this case Brower Fixed Point has an application in game theory specifically in non-cooperative games and existence of Equilibrium. In particular, a game is a set of actions done by the participants defined by a set of rules. This is commonly described using mathematical concepts, which offers a concrete model to describe a variety of situations. On the other hand, the separation axioms Ti, i = 0,1,2,3,4 are vital properties that describes the topological spaces T0,T1,T2,T3 and T4. It is noted that a T3 - space is a generalized version of T2- space and since various results on application of fixed point theory in game theory on an arbitrary locally convex T2 - space has been established, in this study we sort to extend this concept to the general T3 - space. The utilization of a symmetric property of Hausdorff space established that if two continuous commutative mappings are defined on a T3 - space, then the two maps achieves unique fixed points.

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.

Separation Axioms via $F$-open Sets

In this paper, we introduce some types of separation axioms via $F$-open sets, namely $FT_i$ ($i = 0, 1, 2, 3, 4$), $F$-regular and $F$-normal spaces, and investigate their properties, relationships and characterizations. We show that every $FT_i$ space is a Ti space for $i = 0, 1, 2, 3, 4$. However, the converse is true whenever X is finite.

مونويدات جزئية من الزمر الباراتبولوجية الآبيلية

One of the important subclasses of abelian paratopological groups is called the free abelian paratopological group on a topological space. It was introduced in 2003 by Remaguara, Sanchis, and Tkachenko. In this paper, we introduce a single submonoid of the free abelian paratopological group on Alexandroff space, then we prove that this submonoid is a base at the identity element of the free topology of the abelian group. The main result of this paper is to give applications of this submonoid such as studying separation axioms, compactness, and other properties of the free topology on the abelian group. Keywords: free group, Abelian paratopological group, free abelian paratopological group, Alexandroff space, neighborhood base at an identity.

A few separation axioms on Nα*AS - closed sets in nano topological space

This work goal is to gain the relation between a number of the existing sets using inducing and investigating the capabilities of Ɲα*AS -T0 space, Nα*AS-T1space, Nα*AS -T2 space.

WEAKER FORMS ON SEPARATION AXIOMS

In general topology, the concepts of separation axioms play an important role. The main goal of this paper is to present the study of weaker forms of separation axioms called spgwα-T1, spgwα-T2, spgwα-regular and spgwα-normal spaces using spgwα-open sets in topological spaces. Further, the properties spgwα-compact, spgwα-connected and spgwα-Lindelof spaces have been defined and studied their basic characterizations in topological spaces.

On some Separation Axioms in Soft Lattice Topological Spaces

في عالم الطوبولوجيا، يتم فرض قيود مختلفة في كثير من الأحيان على أنواع الفضاءات الطوبولوجية قيد الدراسة ويتم تعريف هذه القيود من خلال ما يعرف ببديهيات الفصل. كما يمكن اعتبار بديهيات الفصل بمثابة شروط إضافية يمكن دمجها في تعريف الفضاءات الطوبولوجية. تخدم بديهيات الفصل أغراضًا مختلفة في نظرية الشبكة. أنها توفر أدوات لتصنيف ومقارنة الشبكات المختلفة، والكشف عن خصائصها الهيكلية والطوبولوجية. في حين أن بديهيات الفصل التقليدية مثل T0، T1، T2، وما إلى ذلك، لا تزال تلعب دورًا، فإن تفسيرها وآثارها تختلف في سياق المجموعات الناعمة والهياكل الشبكية. يقدم هذا البحث بديهيات الفصل، الفضاء T_i للشبكة الناعمة (لـ i = 4، 5، 6)، في سياق الفضاء الطوبولوجي للشبكة الناعمة ويبحث في العديد من الخصائص المرتبطة بها. يمكن لدراسة الشبكات من خلال هذه البديهيات أن تكشف عن الروابط بين خصائصها النظرية وخصائصها الطوبولوجية. يتجاوز هذا العمل مجرد تطبيق بديهيات الفصل على المساحات الطوبولوجية ذات الشبكة الناعمة. إنها تغامر بالكشف عن الخصائص الثابتة للشبكة الناعمة. بالإضافة إلى ذلك، تستكشف الدراسة الخصائص الثابتة للشبكة الناعمة المستمدة من مفاهيم T_i-space للشبكة الناعمة، وعلى وجه التحديد، الخصائص الطوبولوجية للشبكة الناعمة الوراثية والشبكة الناعمة. في الختام، فإن استكشاف الدراسة للخصائص الثابتة للشبكة الناعمة يدفع حدود فهم الفضاءات الطوبولوجية للشبكة الناعمة. ومن خلال الخوض في جوهر هذه الهياكل وخصائصها المحلية والعالمية، تفتح الدراسة الأبواب أمام إمكانيات نظرية مثيرة وتطبيقات محتملة في مجالات متنوعة.

Subgroups of products of Nagata semitopological groups and related results

In this article, we introduce notions which are called property (c*) and property (M 3*) for semitopological groups. We show that if G is a regular semitopological group with a q-point, property (c*) and Sm(G) ≤ ω, then G is topologically isomorphic to a subgroup of the product of a family of first-countable M 1-semitopological groups (Nagata semitopological groups). In the third part of this article, we give an internal characterization of subgroups of product of firstcountable M 1-semitopological groups. A semitopological (paratopological) group G is topologically isomorphic to a subgroup of the product of a family of first-countable M 1-semitopological (paratopological) groups if and only if G satisfies the T 0 separation axiom and has property (M 3*).

Fuzzy SSPO-separation Axioms and Fuzzy α-SSPO Compactness

In this paper we will introduce the concept of new separation axioms named fuzzy SSPO-separation axioms by using the fuzzy strong semi preopen sets and we will also introduce and investigate properties of (alpha)-SSPO compactness. We will define and investigate relation between fuzzy separation axioms, fuzzy pre-separation axioms and different forms of fuzzy continuous mappings. We will also investigate the existence of a countable base of fuzzy strong semi preopen sets, we will define the concept of SSPO separability, the concept of (alpha)-SSPO Lindelof sets and examine their properties. With the concept of fuzzy strong semi pre-continuity, SSPO-irresolute continuous mappings and other forms of fuzzy continuity, we will investigate the new concept of fuzzy compactness and its properties in regards to the mentioned mappings.

Methods of Generating Soft Topologies and Soft Separation Axioms

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies, or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature.

Soft Semi Star Star Generalized Continuous Functions

Recently soft generalized closed sets play important role in studying separation axioms in soft topological spaces. The aim of the present communication is to introduce the concepts of soft semi star star generalized continuous functions and soft semi star star generalized irresolute functions and study their basic properties in soft topological spaces.

New types of nearly open sets in topological spaces and their applications

In the current study, the idea of S ψ φ -open sets is used to introduce some new types of nearly open sets in topological space, where ψ = { θ , δ , τ } and φ = { θ , δ , τ , α , a } , thereby illustrating some of their basic qualities and the links between them. Some characterizations, topological properties, mappings, and separation axioms of these new concepts are also presented.