Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft Open Sets
In this article, new generalised neutrosophic soft open known as neutrosophic soft open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.
1. Introduction and Preliminaries
During the study towards possible applications in classical and nonclassical logic, fuzzy soft sets, vague soft set, and neutrosophic soft set are absolutely important. Nowadays, researchers daily deal with the complexities of modelling uncertain data in economics, engineering, environmental science, sociology, medical science, and many other fields. Classical methods are not always successful due to the reason that uncertainties appearing in these domains may be of various types. Zadeh  originated a new access of fuzzy set theory, which proved to be the most suitable agenda for dealing with uncertainties. While probability theory, rough sets , and other mathematical tools are considered as useful approaches to designate uncertainty. Each of these theories has its own inherent difficulties as pointed out by Molodtsov . Molodtsov  suggested a completely new sophisticated approach of soft sets theory for modelling vagueness and uncertainty which is free from the complications affecting existing methods. In soft set theory the problem of setting the membership function, among other related problems, simply does not arise. Soft sets are considered as neighbourhood systems and are a special case of context-dependent fuzzy sets. Soft set theory has potential applications in many different fields, counting the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory, and measurement theory. Maji et al.  functionalized soft sets in multicriteria decision-making problems by applying the technique of knowledge reduction to the information table induced by the soft set. In , they defined and studied several basic notions of soft set theory. In 2005, Pei and Miao  and Chen  improved the work of Maji et al. Cagman et al.  defined the concept of soft topology on a soft set and presented its related properties. The authors also discussed the foundations of the theory of soft topological spaces. Shabir and Naz  introduced soft topological spaces over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighbourhood of a point, soft separation axioms, and their basic properties were investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. The authors discussed soft subspaces of a soft topological space and investigated characterization with respect to soft open and soft closed sets. Finally, soft -spaces and notions of soft normal and soft regular spaces are addressed. Bayramov and Gunduz  investigated some basic notions of soft topological spaces by using the soft point notion. Later, the authors addressed -soft spaces and the relationships between them. Lastly, the author defined soft compactness and explored some of its significant characteristics.
Khattak et al.  introduced the concept of soft (, ) open sets and their characterization in soft single point topology. Zadeh  introduced the concept of a fuzzy set. The author described that a fuzzy set is a class of objects with a continuum of grades of membership. Further, the authors characterized the set by a membership function, assigning membership grade to each member of the set. The notions of inclusion, union, intersection, complement, relation, convexity, etc. were extended to such sets, and various properties of these notions in the context of fuzzy sets were established. In particular, a separation theorem for convex fuzzy sets was proved ignoring the prerequisites of mutually exclusive fuzzy sets.
Atanassov  introduced the concept of intuitionistic fuzzy set (IFS) which is an extension of the concept “fuzzy set.” The authors discussed various properties including operations and relations over sets. Bayramov and Gunduz  introduced some important properties of intuitionistic fuzzy soft topological spaces and defined the intuitionistic fuzzy soft closure and interior of an intuitionistic fuzzy soft set. Furthermore, intuitionistic fuzzy soft continuous mapping and structural characteristics were discussed in their study. Deli and Broumi  defined for the first define a relation on neutrosophic soft sets. The new concept allows the composition of two neutrosophic soft sets. It is devised to derive useful information through the composition of two neutrosophic soft sets. Finally, a decision-making method on neutrosophic soft sets is presented.
Bera and Mahapatra  introduced the concept of cartesian product and the relationship on neutrosophic soft sets with a new attempt. Some properties of this concept have been discussed and verified with appropriate real-life examples. The neutrosophic soft composition has been defined and verified with the help of examples. Then, some basic properties neutrosophic set have been established. Further, the authors introduced neutrosophic soft function its properties have been introduced and some suitable examples. Injective, surjective, bijective, constant, and identity neutrosophic soft functions have been defined. Finally, properties of inverse neutrosophic soft function have been discussed with proper examples.
Maji  used the concept of the neutrosophic set in the soft set and introduced the neutrosophic soft set. Some definitions and related operations were introduced on the neutrosophic soft set.
Kamal et al.  introduced the concept of multivalued interval neutrosophic soft set which amalgamates multivalued interval neutrosophic set and soft set. According to the authors, the proposed set extended the notions of fuzzy set, intuitionistic fuzzy set, neutrosophic set, interval-valued neutrosophic set, multivalued neutrosophic set, soft set, and neutrosophic soft set. Further, the authors studied some basic operations such as complement, equality, inclusion, union, intersection, “AND”, and “OR” for multivalued interval neutrosophic soft elements and discussed its associated properties. Moreover, the derivation of its properties, related examples, and some proofs on the propositions are included.
Karaaslan and Deli  introduced the concept of soft neutrosophic classical sets and its set theoretical operations such as union, intersection, complement, AND-product, and OR-product. In addition to these concepts and operations, the authors defined four basic types of sets of degenerate elements in a soft neutrosophic classical set. Then, the authors proposed a group decision-making method based on soft neutrosophic classical sets and gave the algorithm of the proposed method. The authors also made an application of the proposed method on a problem including soft neutrosophic classical data.
Karaaslan  studied the concept of single-valued neutrosophic refined soft set is defined as an extension of single-valued neutrosophic refined set. Also, set theoretical operations between two single-valued neutrosophic refined soft sets are defined, and some basic properties of these operations are investigated. Furthermore, two methods to calculate the correlation coefficient between two single-valued neutrosophic refined soft sets are proposed, and a clustering analysis application of one of the proposed methods is given.
Karaaslan  introduced the concept of possibility neutrosophic soft set and defined some related concepts such as possibility neutrosophic soft subset, possibility neutrosophic soft null set, and possibility neutrosophic soft universal set. Then, based on definitions of -norm and -conorm, we define set theoretical operations of possibility neutrosophic soft sets such as union, intersection and complement, and investigate some properties of these operations. The author also introduced AND-product and OR-product operations between two possibility neutrosophic soft sets. The author proposed a decision-making method called the possibility neutrosophic soft decision-making method (PNS decision-making method) which can be applied to the decision-making problems involving uncertainty based on AND-product operation. The author finally gave a numerical example to display the application of the method that can be successfully applied to the problems.
Karaaslan  introduced a similarity measure between possibility neutrosophic soft sets (PNS-set) is defined, and its properties are studied. A decision-making method is established based on the proposed similarity measure. Finally, an application of this similarity measure involving the real-life problem is given.
Karaaslan  addressed firstly Maji’s definitions (Maji-2013) and verified that some propositions are incorrect by a counterexample. The author then redefined the concept of neutrosophic soft set and their operation based on Çağman (Çagman-2014) and investigated some properties of these operations. The author then constructed decision-making method and group decision-making which selects an optimum element from the alternatives by using weights of parameters. The author finally presented an example which shows that the method can be successfully applied to many problems that contain uncertainties.
Bera and Mahapatra  introduced the construction of topology on a neutrosophic soft set (NSS). The notion of the neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, neutrosophic soft boundary, regular NSS, and their basic properties are studied in this study. The base for neutrosophic soft topology and subspace topology on NSS are defined with suitable examples. Some related properties were also established. Moreover, the concept of separation axioms on neutrosophic soft topological space has been introduced along with the investigation of several structural characteristics.
Khattak et al.  for the first time bounced up the idea of the neutrosophic soft open set, neutrosophic soft closed sets and their properties. Also, the idea of neutrosophic soft -neighbourhood and neutrosophic soft -separation axioms in neutrosophic soft topological structures are 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 -separation axioms of neutrosophic soft topological spaces is diffused in different results with respect to soft points. Furthermore, properties of neutrosophic soft b-space () and some associations between them are discussed.
In this article, the concept of vague soft topology is initiated, and its structural characteristics are attempted with new definitions. Some fundamental definitions are given which are necessary for the upcoming study. Neutrosophic soft union, neutrosophic soft intersection, null neutrosophic soft set, absolute neutrosophic soft set, neutrosophic soft topology, and neutrosophic soft neighbourhood are discussed. Three new definitions are introduced with respect to soft points in neutrosophic soft topological spaces. These three definitions are neutrosophic soft semiopen, neutrosophic soft preopen, and neutrosophic soft open sets. Neutrosophic soft open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some neutrosophic soft separation axioms and neutrosophic soft other separation axioms are generated with respect to soft points of the spaces. The interplay among these neutrosophic soft separation axioms is also displayed with respect to soft points of the spaces. These results are verified with examples. The second criteria of space are displayed. The engagement of neutrosophic soft space with open set, neutrosophic soft spaces relation with neutrosophic soft closer with respect to soft points of the spaces.
Neutrosophic soft countability, neutrosophic soft first countability and neutrosophic soft second countability, the relationship between these results, neutrosophic soft sequences and their convergence in Hausdorff space, and the cardinality results are discussed. In continuation, neutrosophic soft topological characteristics and inverse neutrosophic soft topological characteristics are addressed with respect to soft points. Neutrosophic soft product of neutrosophic soft structures is inaugurated. Characterization of Bolzano Weierstrass Property, neutrosophic soft compactness, related results, and neutrosophic sequentially compactness are discussed. In the end, some concluding remarks and future work are planted.
Definition 1 (see ). A neutrosophic set A on the father set is defined as follows: where .
Definition 2 (see ). Let be a father set, be a set of all parameters, and denotes the power set of . A pair is referred to as a soft set over , where is a map given by. Then, a set over is a set defined by a set of valued functions signifying a mapping and is referred to as the approximate set function. It can be written as a set of ordered pairs:
Definition 3 (see ). Let be aNSS over the father set. The complement of is signified and is defined as follows: It is clear that
Definition 4 (see ). Let and two NSS over the father . is supposed to be NSSS of if , , , . It is signified as . is said to be NS equal to if is NSSS of and is NSSS of if . It is symbolized as .
Definition 5 (see ). Let& be two NSSS over father such that . Then, their union is signifies as and is defined as where
Definition 6 (see ). Let and be two NSSS over the fatherset . Then, their intersection is signified as and is defined as
Definition 7 (see ). Let be a NSS over the father set . The complement of is signified and is defined as follows: It is clear that
Definition 8 (see ). NSS be a NSS over the father is said to be a null neutrosophic soft set. If
It is signified as
Definition 9 (see ). NSS over the father set is an absolute NSS if
It is signified as . Clearly, and .
Definition 10 (see ). Let NSS be the family of all NS soft sets and . Then, is said to be a NS soft topology on if (1). (2). The union of any number of NS soft sets in (3). The intersection of a finite number of NS soft sets in . Then, is said to be a NSTS over .
Definition 11 (see ). Let NS be the family of all NS over and then NS is supposed to be a point, for and is defined as follows:
Definition 12 (see ). Let NSS be the family of all soft sets over the father set Then, NSS is called a NS point, for every , and is defined as follows:
Definition 13 (see ). Let be a NSS over the father set We say that read as belonging to the NSS whenever
Definition 14 (see ). be a NSTS over be a over . NSS in is called a NS nbhd of the NS point , if there exists a NS open set such that .
2. Main Results
In this section, three new definitions are introduced with respect to soft points in neutrosophic soft topological spaces. These three definitions are neutrosophic soft semiopen, neutrosophic soft preopen, and neutrosophic soft open sets. Neutrosophic soft open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the help of this new definition, some important results are discussed. Examples are also given for a better understanding of some results. Neutrosophic soft countability that neutrosophic soft first countability and neutrosophic soft second countability, the relationship between these results, neutrosophic soft sequences and their convergence in Hausdorff space, and the cardinality results are discussed.
Definition 15. Let be a NSTS over be a NS set over . Then, is as follows: (1)Neutrosophic soft semiopen if (2)Neutrosophic soft preopen if(3)Neutrosophic soft open if and neutrosophic soft close if
Definition 16. Let be a over , are NS points. If there exist NS open sets such that . Then, is called a NS space.
Definition 17. Let be a NSTS over , are NS points. If there exist NS open sets such that . Then, is called a NSspace.
Definition 18. Let be a NSTS over are NS points. If open set and such that and and . Then, is called a NS space.
Definition 19. Let be a NSTS. be a NS closed set and . If there exists NS open sets and such that and , then is called a NS -regular space. is said to be space, if is both a NS regular and NS space.
Definition 20. Let be a NSTS. This space is a NS normal space, if for every pair of disjoint NS close sets and disjoint NS open sets and
is said to be a NS space if it is both a NS normal and NS space.
Example 21. Suppose that the father set is assumed to be and the set of conditions by. Let us consider set and , , , and be points. Then, the family , where , is a . Thus, (,, ) be a . Also, is structure but it is not because for NS points not
Example 22. Suppose that the father set is assumed to be and the set of conditions by . Let us consider set and , , , and be points. Then, the family is a Thus, be a. Also, is structure.
Theorem 23. Let be a . Then, be a structure if each point is a -close.
Proof. Let be a over be an arbitrary point. We establish is a soft open set. Let Then, either . This means that are two are distinct Thus, or or or or or or or . Since be a structure, a open set so that and Since, . So Thus, is a open set, i.e., is a closed set. Suppose that each point is a closed set. Then, is a open set. Let Thus, and . So be a - space.☐
Theorem 24. Let be a over the father Then, is - space if and only if for distinct NS points and , there exists a open set containing there exists but not such that
Proof. Let be two points in space. Then, there exist disjoint open sets such that . Since and Next, suppose that , there exists a open set containing but not that is are mutually exclusive open sets supposing in turn.☐
Theorem 25. Let be a Then, is space if every point . If there exists a open set such that then a space.
Proof. Suppose. Since is space. and are closed setsin . Then, . Thus, there exists a open set such that So we have and , that is is a soft space.☐
Theorem 26. Let be a is soft space if for every that is (such that
Proof. Let is space and Since is space for the NS point and closed set, and such that and . Then, we have Since closed set. Conversely, let be a closed set. and from the condition of the theorem, we have. Thus, So is space.☐
Theorem 27. Let (,, ) be a over the father set This space is a space, if and only if for each closed set and open set with , there exists a open set .
Proof. Let be a over the father set and let . Then, is a closed set and Since ,, ) be a space, open sets Thus, is a closed set and . So
Conversely, and be two disjoint closed sets. Then, implies open set . Thus, are open sets and , and and (,, ) be a space.☐
Theorem 28. Let be a point in a first countable space and let generates a countable open base about the point ; then, there exists an infinite soft subsequence of the sequence , such that (i) for any open set , containing , a suffix such that for all and (ii) if be, in particular, a -space, then
Proof. Given is open sets, containing the point . As the sets form open base about , there exists one among the sets , which we shall denote by , such that (