Soft topology

The aim of this study is to introduce and investigate two new classes of separation axioms called supra soft R 0 and supra soft R 1 . They are defined in the spaces of supra soft topologies by using the notions of supra soft open sets and supra soft closure operator. We discuss the basic properties and characterizations of them. We also study the relationships between these classes and some other supra soft separation axioms with many results and explanative examples. Moreover, the connections between the properties of these classes and those in some generated soft topologies are presented. Finally, we show that these classes are preserved under subspaces, which means they are supra soft topological properties.

In this paper, we introduce the concepts of 𝐷 𝛼 𝑝-open, 𝐷 𝛼 𝑝-closed subsets, pairwise-𝛼-closed, pairwise-𝑔-closed subsets, pairwise-strongly 𝛼-closed graph 𝐺(𝑓) and strongly 𝐷 𝛼 𝑝-closed graph of bitopological spaces. We showed that each closed graph is 𝐷 𝛼 𝑝-closed. In addition, the concepts of 𝐷 𝛼 𝑝-continuous, open, and closed functions are defined, and the relations between 𝜏 𝑝-𝛼, 𝜏 𝑝-𝑔, and 𝐷 𝛼 𝑝-continuous functions are clarified. The fact that strongly 𝐷 𝛼 𝑝-closed graph is 𝐷 𝛼 𝑝-closed is illustrated. We studied when the graph 𝐺(𝑓) is 𝑝-strongly closed and 𝑝-𝐷 𝛼-closed subsets of the bitopological space (X, τ₁, τ₂). Moreover, the notions of 𝐷 𝛼 𝑖-interior of a subset of 𝑋 and 𝐷 𝛼 𝑖-closure are defined.

This work presents a generalized notion of multiset mapping thus resolving a long standing obstacle in structural study of multiset processing. It has been shown that the mapping defined herein can model a vast array of notions as special cases and also handels diverse situations in multiset rewriting transformations. Specifically, this paper unifies and generalizes the works of Parikh(1966), Hickman(1980), Khomenko(2003) and Nazmul(2013).

Topics in Generalized Topology and Fuzzy Generalized Continuities This work is comprised of the generalized topology, algebraic generalized topology and fuzzy generalized topology. We have defined and studied the notions of semi-local function, global function, global closure operator, global interior and-normal spaces in Ideal generalized topological spaces. Properties of these functions are investigated. Examples and counter examples are given where deemed necessary. We continued the investigations of some important results in generalized topological groups and proved basic properties of-connectedness. Along with other results, it is proved that in a-topological group, the maximal-connected components containing the identity of the group is-closed invariant subgroup.-quotients of-topological groups are also discussed.-topological vector spaces (generalization of TVS) are one of the interesting structures which is defined and investigated over here. This space is a blend of-topological structure with the algebraic concept of a vector space in such a way that vector addition and scalar multiplication are-continuous functions. A counter example is given to show that a-topological vector space is not a topological vector space. We have studied the concept of a fuzzy generalized topology which is a generalization of Chang's fuzzy topology, and investigated some properties of its structure. We have also introduced the concept of fuzzy generalized open function and fuzzy generalized closed function in terms of fuzzy generalized interior and fuzzy generalized closure operators 9 2.1-Topological Spaces .2 Ideal-Topological Spaces 16 .3-semi compatible with an ideal .4 lobal functions Ideal-Topological Spaces .5 lobal closure and Global interior 34-normal spaces 38 .7-regular spaces 3-Topological Groups 47 .

In this paper we introduce a new class of functions called -quasi irresolute functions. The notion of -quasi graphs are introduced and the relationship between -quasi irresolute functions and -quasi closed graphs is analysed. 2000 Mathematics Subject Classification : 54C08, 54C10.

The main goal of this paper is to introduce and study two new class of functions, called weakly e *-irresolute functions and strongly e *-irresolute functions, via the notion of e *open set defined by Ekici [7]. We obtain several fundamental properties and characterizations of these functions. Moreover, we investigate not only some of their basic properties but also their relationships with other types of already existing topological functions.

Herein, we define fuzzy T 0-space, fuzzy T 1-space, fuzzy T 2 (or Hausdorff), as well as fuzzy regular and fuzzy normal spaces in mixed fuzzy topological spaces, and then establish relationships among these spaces. We provide some results for the abovementioned spaces in mixed fuzzy topological spaces.

In the present paper, soft-open and soft-closed sets in soft topological spaces are defined over an initial universe with a set of parameters. A necessary and sufficient condition for a soft set to be soft-open set in soft topological space is stated and proved. A detailed study is carried out on properties of soft-interier and soft-closure of soft sets.

In this article semi-open and semi-closed sets on a fuzzy soft topological space are introduced and some properties have been studied. Fuzzy soft semi closure and fuzzy soft semi interior operators are also introduced and studied. Semi-continuous, semi-open, semi-clsoed and irresolute fuzzy soft functions on a fuzzy soft topological space are introduced and characterized.

In this paper, we introduce the notions of pairwise soft sub kernel, and pairwise ∨-soft sets in a soft bitopological space (X, η 1 , η 2 , E). Also, we study the fundamental properties of pairwise ∨-soft sets and we investigate the associated soft topology η p∨. Moreover, we introduce the notions of generalized pairwise (∨)-soft sets and we investigate their basic properties. Also, we define a soft closure operator on the family of all generalized pairwise-soft sets and generate, in usual manner, an Alexandroff soft topology η gp on X which is finer than η p∨. Furthermore, we prove that (X, η gp , E) is always a soft T 1 2 space. In addition, we introduce characterizations of pairwise soft T * 1 2 by using generalized pairwise-soft sets and we show that the concepts of generalized pairwise-soft set and generalized pairwise closed soft set are independent.

Kandil and his colleagues [10], introduced the notion of λ p-closed soft set by involving Λ p-soft set and pclosed soft set. In this paper, we give some additional properties of λ p-closed soft sets. We also introduce and study a related new class of 1 4 � *-spaces which lies between * and * λ PST. We provide some illustrative examples to support the results.

The aim of this study is to introduce and investigate two new classes of separation axioms called supra soft R 0 and supra soft R 1. They are defined in the spaces of supra soft topologies by using the notions of supra soft open sets and supra soft closure operator. We discuss the basic properties and characterizations of them. We also study the relationships between these classes and some other supra soft separation axioms with many results and explanative examples. Moreover, the connections between the properties of these classes and those in some generated soft topologies are presented. Finally, we show that these classes are preserved under subspaces, which means they are supra soft topological properties.

Nakaoka and Oda ([1] and [2]) initiated the notion of maximal open (resp. minimal closed) sets in topological spaces. Thereafter, in 2005, Cao,Ganster, Reilly and Steiner [4] introduced delta thita-open (resp. delta thita-closed) sets in general topology. In the present work, the author introduces new classes of open and closed sets called maximal delta thita-open sets, minimal delta thita-open sets, maximal delta thita-closed sets, minimal delta thita-closed sets, delta thita-semi maximal open and delta thita-semi minimal closed and investigate some of their fundamental properties.

In this paper, we introduce and study some new soft properties namely, soft R0 and soft R1(SRi, for short i = 0, 1) by using the concept of distinct soft points and we obtain some of their properties. We show how they relate to some soft separation axioms in [21]. Also we, show that the properties SR0, SR1 are special cases of soft regularity. We further, show that in the case of soft compact spaces, SR1 is equivalent to soft regularity. Finally, the relations between these properties in soft topologies and that in crisp topologies are studied. Moreover, some counterexamples are given.

This work aims to introduce and discuss two new classes of separation properties namely, soft generalized R 0 and R 1 in a soft generalized topological space defined on an initial universe set, by using the notions of soft g-open sets and soft gclosure operator. We investigate some of their properties and characterizations. We further, investigate the relationships between different generalized structures of soft topology, providing some illustrative examples and results. Additionally, we present connections between these separation properties and those in some generated topologies. Furthermore, we show that being SGR i , i = 0, 1 are soft generalized topological properties.

The aim of this paper is to introduce and investigate a new class of functions called weakly almost contra-T *-continuity which is defined as a function from an operator topological space (X, τ, T) into an arbitrary topological space (Y, δ). Furthermore, some new characterizations, several basic propositions are proved and some relevant counterexamples are provided.

The aim of this paper is to introduce and study $\mathcal{B}$-open sets and related properties. Also, we define a bi-operator topological space $(X, \tau, T_1, T_2)$, involving the two operators $T_1$ and $T_2$, which are used to define $\mathcal{B}$-open sets. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is, in turn, a generalization of a pre-open set and a semi-open set. We introduce a number of concepts based on $\mathcal{B}$-open sets.

A new class of functions, called slightly generalized β-continuous functions is introduced. Basic properties of slightly generalized β-continuous functions are studied. The class of slightly generalized βcontinuous functions properly includes the class of slightly β-continuous functions and generalized β-continuous functions. Also, by using slightly generalized β-continuous functions, some properties of domain/range of functions are characterized. 2010 Mathematics Subject Classification. 54C08.

Kandil and his colleagues [10], introduced the notion of -closed soft set by involving -soft set and -closed soft set. In this paper, we give some additional properties of -closed soft sets. We also introduce and study a related new class of -spaces which lies between  and  . Moreover, we show that there exists a very important relation between the notion of -closed soft sets and the  property, ,  , . In addition, we offer the notion of -locally closed soft sets and we investigate a related new pairwise soft separation axiom  which is independent from . The relationships between the -closed soft sets and the -locally closed soft sets are obtained. Furthermore, we introduce the notion of -open soft sets and we construct supra soft topology associated with the class of -open soft sets and we present pairwise soft separation axioms related to such soft sets, namely . We provide some illustrative examples to support the results.

The main goal of this paper is to introduce and study a new type of contra continuity called contra we *-continuity via we *-open sets defined by us. We obtain several characterizations and some fundamental properties of contra we *-continuous functions. Moreover, we investigate the relationships between contra we *-continuous functions and other related strong and weak forms of contra continuity. Furthermore, we introduce the concepts of we *-compactness and strongly contra we *-closed graphs to look into some different properties.

In soft topological space there are some existing related concepts such as soft open, soft closed, soft subspace, soft separation axioms, soft connectedness, soft locally connectedness. In this paper, a new class of soft sets called maximal soft δ-open sets and minimal soft δ-closed sets which are fundamental results for further research are defined on soft topological space and continued in investigating the properties of these new notions of open sets with example and counter examples.

The author introduces a new separation axiom and studies so me of their basic properties. The implication of these new separation axiom among themselves and with the well known axioms semi-T 2 semi-T 1 and semi-T 0 are obtained. l. Introduction 1 Semi-open sets where introduced and investigated by N. Levine [6] in 1963. In 1975, S.N. Maheshwari and R. Prasad [7] used semi-open sets to define and investigate three new separation axioms, called semi-T2o semi-T 1 and semi-T 0. Moreover, they have shown that the following implications hold.

Many mathematicians defined and studied soft separation axioms and soft continuity in soft spaces by using ordinary points of a topological space X. Also, some of them studied the same concepts by using soft points. In this paper, we introduce the concepts of soft p c − T i {p}_{c}-{T}_{i} and soft p c − T i ⁎ {p}_{c}-{T}_{i}^{\ast } , i = 0 , 1 , 2 i=0,1,2 by using the concept of soft p c {p}_{c} -open sets in soft topological spaces. We explore several properties of such spaces. We also investigate the relationship among these spaces and provide a counter example when it is needed.

The concept of (b, s)-continuous functions in topological spaces is introduced and studied. Some of their characteristic properties are considered. Also we investigate the relationships between these classes of functions and other classes of functions. 1991 Math. Subject Classification. 54C10, 54D10.

In this paper, regularity and separation axioms in fuzzy soft topological spaces are defined and studied by using quasi-coincident relation and fuzzy soft neighborhood system. We discuss its charaterizations and relationship among them. In addition, goodness and hereditary properties are discussed.

In this paper, we are going to introduce an approach to convert the topology of a topological space to another topology (in fact, a coarser topology). For this purpose, we consider a topological space (X, τ) with a closed point p of its points. Using a grill G on the space (X, τ) and the closure operator associated with τ , we define a Kuratowski closure operator cl * p on X to create the desired topology. We then characterize the form of this resulting topology and also determine its relationship to the initial topology of the space.

The aim of this paper is to introduce and investigate a new class of functions called weakly almost contra-T *-continuity which is defined as a function from an operator topological space (X, τ, T) into an arbitrary topological space (Y, δ). Furthermore, some new characterizations, several basic propositions are proved and some relevant counterexamples are provided.

In this paper, we introduce the notations of e-I-open sets and strong B * I-set to obtain a decomposition of continuing via idealization. Additionally, we investigate properties of e-I-open sets and strong B * Iset. Also we studied some more properties of e-I-open sets and obtained several characterizations of e-I-continuous functions and investigate their relationship with other types of functions.

In this paper, the notion of fuzzy resolvability modulo ideal, with respect to the fuzzy local function in a fuzzy topological space, was investigated. A trial was made to explain the notions of resolvability of complementary fuzzy dense set, fuzzy local closed set and fuzzy I-condense set.

In this paper, we further the study of -compactness a general- ization of quasi-H-closed sets and its applications to some forms of continuity using -open and -open sets. Among other results, it is shown a weakly -retract of a Hausdor space X is a -closed subset of X.

This paper is devoted to introduce new definitions of fuzzy αopen set and new weaker forms of fuzzy αcontinuous mappings. Properties and relationship between fuzzy αopen sets and fuzzy αcontinuous mappings and other weaker form of them are investigated.

The aim of this paper is to introduce the new class of intuitionistic fuzzy closed sets called intuitionistic fuzzy rw-closed sets in ituitionistic fuzzy topological space. The class of all intuitionistic fuzzy rw-closed sets lies between the class of all intuitionistic fuzzy w-closed sets and class of all intuitionistic fuzzy rg-closed sets. We also introduce the concepts of intuitionistic fuzzy rw-open sets, intuitionistic fuzzy rw-continuity and intuitionistic fuzzy rw-open and intuitionistic fuzzy rw-closed mappings in intuitionistic fuzzy topological spaces. Keywords: Intuitionistic fuzzy rw-closed sets, Intuitionistic fuzzy rw-open sets, Intuitionistic fuzzy rw-connectedness, Intuitionistic fuzzy rw-compactness, intuitionistic fuzzy rw-continuous mappings, Intuitionistic fuzzy rw-open mappings and intuitionistic fuzzy rw-closed mappings.

The aim of this paper is to introduce two new classes of functions, namely slightly *-continuous functions and totally *-continuous functions and study its properties. Mathematics Subject Classification: 54A05 Keywords and phrases: *-open set, *-closed set, slightly *-continuous functions and totally *-continuous functions. * (A) denote the *-closure and the *-interior of A respectively. Definition 2.1: A subset A of a topological space (X,τ) is called a *-open set [6] if there is an open set U in X such that U⊆A⊆ sCl * (U). The collection of all S g *-open sets in (X, τ) is denoted by * O(X,τ). Definition 2.2: A subset A of a topological space (X,τ) is called a *-closed set[6] if X\A is *-open. The collection of all *-closed sets in (X, τ) is denoted by * (X,τ). Theorem 2.3 [6]:Every open set is S g *-open and every closed set is S g *-closed set Definition 2.4: A topological space X, τ is said to be *-space [7] if every *-open set of X is open in X. Definition 2.5: A topological space X, τ is said to be *-locally indiscrete space [8] if every *-open set of X is closed in X. Definition 2.6: A function f: X → Y is said to be contra-*-continuous [8] if the inverse image of every open set in Y is *-closed in X. Definition 2.7: A function f: (X, τ) ⟶ (Y, σ) is called a contra-continuous [2] if f −1 (O) is closed in (X, τ) for every open subset O of (Y, σ). Definition 2.8: A mapping f: X → Y is said to be *-continuous [7] if the inverse image of every open set in Y is *-open in X. Defintion 2.9: A map f: X → Y is said to be *-irresolute[7] if the inverse image of every S g *-open set in Y is *-open in X. Definition 2.10: A mapping f: X → Y is said to be strongly *-continuous [7]if the inverse image of every *open set in Y is open in X. Definition 2.11: A mapping f: X → Y is said to be perfectly *-continuous [7]if the inverse image of every *open set in Y is open and closed in X. Slightly *-continuous functions and Totally *-continuous functions in Topological spaces

In this manuscript, the concept of a generalized fuzzy soft point is introduced and some of its basic properties were studied. Also, the concepts of a generalized fuzzy soft base (subbase) and a generalized fuzzy soft subspace were introduced and some important theorems were established. Finally, the relationship between fuzzy soft set, intuitionistic fuzzy soft set, generalized fuzzy soft set and generalized intuitionistic fuzzy soft set were investigated.

In this paper we intend to introduce a new class of sets known as e-I+-open sets, defined in the light of simple extension topology and ideal topology. This set is investigated and found to be a weaker form of e-I-open sets. We have also generalized this concept and studied its properties.

This paper deals with some structural properties of fuzzy soft topological spaces. Fuzzy soft closure and fuzzy soft interior of a fuzzy soft set are studied and investigated. Fuzzy soft exterior and fuzzy soft boundary of a fuzzy soft set are introduced and some properties related to these structures are established.

The idea of neutrosophic set was floated by Smarandache by supposing a truth membership, an indeterminacy membership and a falsehood or falsity membership functions. Neutrosophic soft sets bonded by Maji have been utilized successfully to model uncertainty in several areas of application such as control, reasoning, pattern recognition and computer vision. The first aim of this article bounces the idea of neutrosophic soft b-open set, neutrosophic soft b-closed sets and their properties. Also the idea of neutrosophic soft b-neighborhood and neutrosophic soft b-separation axioms in neutrosophic soft topological structures are also reflected here. Later on the important results are discussed related to these newly defined concepts with respect to soft points. The concept of neutrosophic soft b-separation axioms of neutrosophic soft topological spaces is diffused in different results with respect to soft points. Furthermore, properties of neutrosophic soft b-T i-space (i = 0, 1, 2, 3, 4) and some linkage between them are discussed.

The aim of this paper is to define and study B-open sets and related properties. A B-open set is, roughly speaking, a generalization of a b-open set, which is in turn a generalization of a pre-open set and a semi-open set. Using B-open sets, we introduce a number of concepts such as B-dense, B-Frechet, contra-B-closed graph and contra-B-continuity. Also, we define a bi-operator topological space (X, τ, T 1 , T 2) which involves two operators T 1 and T 2 , which are used to define B-open sets.

In this paper, union, and intersection of generalised fuzzy soft sets are introduced and some of their basic properties are studied. The objective of this paper is to introduce the generalised fuzzy soft topology over a soft universe with a fixed set of parameters. Generalised fuzzy soft points, generalised fuzzy soft closure, generalised fuzzy soft neighbourhood, generalised fuzzy soft interior, generalised fuzzy soft base are introduced and their basic properties are investigated. Finally, generalised fuzzy soft compact spaces are introduced and a few basic properties are taken up for consideration.

In this paper, the notion of fuzzy resolvability modulo ideal, with respect to the fuzzy local function in a fuzzy topological space, was investigated. A trial was made to explain the notions of resolvability of complementary fuzzy dense set, fuzzy local closed set and fuzzy I-condense set.

In 1999, Molodtsov initiated the concept of Soft Sets Theory as a new mathematical tool and a completely different approach for dealing with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied the theory of soft topological spaces, also defining and investigating many new soft properties as generalization of the classical ones. In this paper, we introduce the notions of soft separation between soft points and soft closed sets in order to obtain a generalization of the well-known Embedding Lemma for soft topological spaces.