Separation Axioms Research Articles

Separation Axioms on Bipolar Hypersoft Topological Spaces

According to its definition, a topological space could be a highly unexpected object. There are spaces (indiscreet space) which have only two open sets: the empty set and the entire space. In a discrete space, on the other hand, each set is open. These two artificial extremes are very rarely seen in actual practice. Most spaces in geometry and analysis fall somewhere between these two types of spaces. Accordingly, the separation axioms allow us to say with confidence whether a topological space contains a sufficient number of open sets to meet our needs. To this end, we use bipolar hypersoft (BHS) sets (one of the efficient tools to deal with ambiguity and vagueness) to define a new kind of separation axioms called BHS Ti-space (i = 0, 1, 2, 3, 4). We show that ee BHS Ti-space (i = 1,2) implies BHS Ti−1-space; however, the converse is false, as shown by an example. e For i = 0, 1, 2, 3, 4, we prove that BHS Ti -space is hypersoft (HS) Ti -space and we present a condition so that eee HS Ti-space is BHS Ti-space. Moreover, we study that a BHS subspace of a BHS Ti-space is a BHS Ti-space for i = 0,1,2,3.

Ic-Separation Axioms in Topological Spaces

A new class of separation axioms known as ic-separation axioms is introduced in this study. They rely on new extended open sets known as ic-open sets, and we describe the relationship between them and give several examples. Additionally, we define ic-generalized closed set concepts in a topological space in order to frame the another class of separation axioms called ic-generalized axioms. Among other things, the basic concern properties and relative preservation properties of these spaces are projected under ic-generalized irresolute mappings.

Pre-separation Axioms in Neutrosophic Topological Spaces

In this article, we first establish a few relationships among neutrosophic interior, neutrosophic closure, neutrosophic pre-open sets, and neutrosophic pre-closed sets in single-valued neutrosophic topological spaces. Thereafter, we defined neutrosophic pre- space, neutrosophic pre- space, and neutrosophic pre- space based on single-valued neutrosophic topological spaces and studied a few properties and relationships among them. We try to establish some relationships between existing neutrosophic separation axioms and newly defined neutrosophic pre-separation axioms. Finally, we study some hereditary properties of pre-separation axioms. Apart from these, we also explore some results implementing neutrosophic pre-open function, neutrosophic pre-continuous function, neutrosophic pre-irresolute function and neutrosophic pre -function based on our defined definitions.

APPROXIMATION SPACES VIA IDEALS AND GRILLS

In this paper, we use the notions of lower set $L_{R}(A)$ and the upper set $U_{R}(A)$ to define the interior operator ${\rm int}_{R}^{A}$ and the closure operator ${\rm cl}_{R}^{A}$ associated with a set $A$ in an approximation space $(X,R)$. These operators generate an approximation topological space different from the generated Nano topological space in $(X,R)$. Ideal approximation spaces $(X,R, \ell)$ based on an ideal $\ell$ joined to the approximation space $(X,R)$ are introduced as well. The approximation continuity and the ideal approximation continuity are defined. The lower separation axioms $T_{i}, i= 0,1,2$ are introduced in the approximation spaces and also in the ideal approximation spaces. Examples are given to explain the definitions. Connectedness in approximation spaces and ideal connectedness are introduced and the differences between them are explained. The interior and the closure operators are deduced using a grill ${\cal G}$ defined on $(X,R)$, yielding the same results.

Some separation axioms for partially ordered sets

The main goal of this paper is to introduce and study new classes of partial orderings on a set with Krull dimension at most 1. We show that these classes are related to door, submaximal and Whyburn T0-spaces.

On β*-supra topological spaces

In this paper, a new type of supra closed sets is introduced which we called supra β*-closed sets in a supra topological space. A new set of separation axioms is defined, and its many properties are examined. The relationships between supra β*-Ti –spaces (i = 0, 1, 2) are studied and shown with instances. Additionally, new varieties of supra β*-continuous maps have been taken into consideration based on the supra β*-open sets theory.

LOW SEPARATION AXIOMS ASSOCIATED WITH g^∗s-CLOSED SETS

LOW SEPARATION AXIOMS ASSOCIATED WITH g^∗s-CLOSED SETS

Continuity and separation axioms via infra-topological spaces

In order to investigate a particular topic in mathematics, more specifically, general topology, it is always desirable to find a weaker condition. This work is planned to study a weak (topological) structure named infra-topological space. An infra-topological space is the collection of subsets of a universe that includes the empty set and is closed under finite intersections. The continuity, openness, and homeomorphism of mappings between infra-topological spaces are explored. Through the use of some examples, analogous properties and characterizations of ordinary mappings cannot be hopped on infra-topological structures. Then, the concepts of product and coproduct of infra-topological spaces are analyzed. Furthermore, the notion of infra-quotient topologies, which are inspired by infra-continuity, is introduced. The essential properties indicate that infra-quotient topologies and ordinary quotient topologies act in parallel. The final part of this paper is devoted to the investigation of infra separation axioms (infra \(T_i\)-spaces, \(i=0,1,\ldots, 4\)). The behaviour of ordinary separation axioms cannot be translated to an infra-topological structure. More precisely, infra-\(T_3\) and infra-\(T_4\)-spaces are independent, and singletons need not be infra-closed in infra-\(T_1\)-spaces.

Hausdorff of a cycle in topological graph

The main purpose of the paper is to bring together two areas in which strong relation, graph theory and topological spaces. And derive interesting formula for the set that contains all minimal dominating sets(MDS) and ( -set). Some separation axioms are discussed in topological graph theory, especially in a cycle graph Cn.

β*-Regular supra topological spaces

Form the series of generalization of the topic of supra topology is the generalization of separation axioms . In this paper we have been introduced (S * - SS *) regular spaces . Most of the properties of both spaces have been investigated and reinforced with examples . In the last part we presented the notations of supra *- -space ( =0,1) and we studied their relationship with (S * - SS *) regular spaces.

Some Result on Supra Separation Axioms via Graph Theory

The Concepts of supra topological graph and introduces of class supra separation axioms on supra topological graph (supra gT0, supra gT1,supra gT2,supra gT3,supra gT4), many relations among them were studied. Gave results for them.

Applications of Nano Penta (N_P) Separation axioms using N_P_ open sets

The main goal of this paper is to use the concept _ open sets to present new classes of separation axioms in _topological spaces. Those new classes are _spaces, =0,1,2. We have studied some basic properties of these spaces. We also discussed the relationship between _spaces and Nano separation axioms spaces), =0,1,2. Furthermore the paper deals with the relationship between the separation axioms throughout kernel set associated with the closed set which used to prove some theorems related to it. The hereditary and topological properties were also discussed.

Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms

This paper contributes to the field of supra-soft topology. We introduce and investigate supra pp-soft Tj and supra pt-soft Tj-spaces (j=0,1,2,3,4). These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra tp-soft Tj and supra tt-soft Tj-spaces (j=0,1,2,3,4). This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for i=0,1, possessing the Ti property by a parametric supra-topological space implies possessing the pp-soft Ti property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of pp-soft Ti-spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft S

Some Study on the Topological Structure on Semigroups

Some studies related to the topological structure of semigroups are provided. In, [3], considering and investigating the properties of the collection A of all the proper uniformly strongly prime ideals of a Γ - semigroup S, such study starts by constructing a topology τA on A using a closure operator defined in terms of the intersection and inclusion relation among these ideals of Γ-semigroup S, which is a generalization of the semigroup. In this paper, we introduce three other classes of ideals in semigroups called maximal ideals, prime ideals and strongly irreducible ideals, respectively. Investigating properties of the collection M, B and S of all proper maximal ideals, prime ideals and strongly irreducible ideals, respectively, of a semigroup S, we construct the respective topologies on them. The respective obtained topological spaces are called the structure spaces of the semigroup S. We study several principal topological axioms and properties in those structure spaces of semigroup such as separation axioms, compactness and connectedness, etc.

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.

On separation axioms of topological rough groups

On separation axioms of topological rough groups

On Some New Concepts of Weakly Neutrosophic Crisp Separation Axioms

On Some New Concepts of Weakly Neutrosophic Crisp Separation Axioms

New Types of Separation Axioms via Wp-Open Sets

In this work, new kinds of separation axioms using wp open sets, some results, properties, examples, and the relationship between these concepts have been given to support our work.

New Concepts of To Separation Axioms in Fuzzy Topology

Fuzzy set was introduced by Zadeh in his classical paper of 1965. Three years later, Chang gave the definition of fuzzy topology, which is a family of fuzzy set satisfying the three classical axioms. In this paper, we have introduced and studied some new notions of To separation axioms in fuzzy topological spaces by using quasi-coincident relation for fuzzy set. Every ordinary (crisp) topological space vacuously satisfies condition of being quasi-To. In this paper concept of quasi coincident relation used to introduce and investigate some quasi separation axioms such as To, T1, & T2. Concerning quasi-To space in the general frame work of fuzzy topological spaces.

Sets, Properties and Truth Values: A Category-Theoretic Approach to Zermelo’s Axiom of Separation

In 1908 the German mathematician Ernst Zermelo gave an axiomatization of the concept of set. His axioms remain at the core of what became to be known as Zermelo-Fraenkel set theory. There were two axioms that received diverse criticisms at the time: the axiom of choice and the axiom of separation. This paper centers around one question this latter axiom raised. The main purpose is to show how this question might be solved with the aid of another, more recent mathematical theory of sets which, like Zermelo’s, has numerous philosophical underpinnings. Keywords: properties of sets, foundations of mathematics, axiom of separation, subobject classifier, truth values