Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft Open Sets

Abstract

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 [1] 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 [2], 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 [3]. Molodtsov [4] 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. [5] 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 [6], they defined and studied several basic notions of soft set theory. In 2005, Pei and Miao [7] and Chen [8] improved the work of Maji et al. Cagman et al. [9] 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 [10] 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 [11] 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. [12] introduced the concept of soft (, ) open sets and their characterization in soft single point topology. Zadeh [1] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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. [18] 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 [19] 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 [20] 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 [21] 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 [22] 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 [23] 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 [24] 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. [25] 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 [25]). A neutrosophic set A on the father set is defined as follows: where .

Definition 2 (see [15]). 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 [24]). Let be aNSS over the father set. The complement of is signified and is defined as follows: It is clear that

Definition 4 (see [17]). 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 [25]). Let& be two NSSS over father such that . Then, their union is signifies as and is defined as where

Definition 6 (see [25]). Let and be two NSSS over the fatherset . Then, their intersection is signified as and is defined as
where

Definition 7 (see [25]). 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 [25]). NSS be a NSS over the father is said to be a null neutrosophic soft set. If

It is signified as

Definition 9 (see [25]). NSS over the father set is an absolute NSS if

It is signified as . Clearly, and .

Definition 10 (see [25]). 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 [25]). 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 [25]). 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 [25]). Let be a NSS over the father set We say that read as belonging to the NSS whenever

Definition 14 (see [25]). 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 (. The sequence thus obtained, has the required properties. In fact, if is any open set, containing , then a set say, belonging to the family , such that . Also, since . Next, let be -space, and let. As is contained in each that is it follows that . Let be any point of from that is or. By definition of -space, there exists open set such that and . There exists a suffix, Consequently, hence, . Thus, consists of the point only.

Theorem 29. second count-ability is first count-ability.

Proof. Let be a soft 2nd countable space. Then, this situation permits that there lives a countable base for . In order to justify that is , we proceed as, let be an arbitrary point. Let us assemble those members of which absorbs and named as . If is soft -hood of , then this permit open set arresting , in and so in such that . This justifies that is a local base at . One step more, being a subfamily of a countable family , it is therefore countable. Thus, every crisp point of supposes a countable local base. This leads us to say is soft first countable.

Theorem 30. A countable space in which every convergent sequence has a unique soft limit is a Hausdorff space.

Proof. Let be Hausdorff space and let be a soft convergent sequence in . We prove that the limit of this sequence is unique. We prove this result by contradiction. Suppose converges to two soft points and such that . Then, by trichotomy law either or. Since it possess the Hausdorff characteristics, there must happen two open sets such that .
Now, converges to so there exists an integer such that . Also, converges to so there exists an integer such that . We are interested todiscuss the maximum possibility, for that we must suppose the maximum of both the integers which will enable us to discuss the soft sequence for a single soft number. , which leads to the situation and . This implies that and . This guarantees that , which beautifully contradict the fact that Hence, the limit of the sequence must be unique.

Theorem 31. The cardinality of all open sets in a second countable space is at most equal to (the power of the continuum).

Proof. Let be such that is second countable space. Let be any soft set of , then is the soft union of a certain soft subcollection of the countable collection . Hence, the cardinality of the set of all soft open sets in is not greater than the cardinality of the soft set of all soft subcollections of the countable collection Thus, the cardinality of .

Theorem 32. Any collection of mutually exclusive open sets in a second countable space is at most countable.

Proof. Let be such that it is second countable. Let signifies any collection of mutually exclusive open sets in Let then at least one soft set , belonging to the collection , such that Let be the smallest suffix for which Since the soft sets in are mutually disjoint, it follows that, for different soft sets there correspond soft sets with different suffices Hence, the soft sets in are in a -correspondence with a soft subcollection of the countable collection ; consequently, the cardinality of is less than or equal to

Theorem 33. Let be such that it is second countabley Hausdorff space. Then, set of all open sets has the cardinality .

Proof. Let be second countable Hausdorff space. There exists in an infinite soft sequence of open sets such that
. Different soft subsequences of the sequences will determine, as their unions, different open sets. But since the soft set of all soft subsets of a countable soft set has the cardinality , it follows that soft set of all the open sets in has the cardinality . Again, the cardinality of the soft set of all open sets in Consequently, the cardinality of all open sets in is exactly equal to

Theorem 34. Let be NSST such that it is second countable space; the soft set of all points in this space has the cardinality at most equal to (i.e., possess the power of the continuum at most).

Proof. Given that is second countable space, for which the sets Forms a soft countable open base of Then, for any given point generate countable open base about the point in ,where is a soft subsequence of the soft sequence , consisting of all those which contain the point . Since a second countable space is necessary first countable. So corresponding to the point , there exists an infinite soft sequence of open sets , which is a soft subsequence of the soft sequence and, therefore, also a soft subsequence of the soft sequence , such that . Thus, to each point in , there corresponds a soft subsequence of the soft sequence , and two in different points, there correspond two such different soft subspaces . Hence, the soft set of all points in , i.e., the crisp set , has the same cardinality as that of a certain soft subcollection of the soft collection of all soft subspace of the sequence . Thus, the cardinality of is less than or equal to In other words, the crisp set has the power of the continuum at most.

Theorem 35. Let be such that it is second countable. Every soft uncountable soft subset contains a point of condensation.

Proof. Since is second countable space and be a soft countable open base of . Let be a subset of such that does not contain any point of condensation. For each point , is not the point of condensation of . Hence, there exists open set , containing such that is soft countable at most. There exists a suffix such that , and then, is also countable at most. But we can express in the form and there can be at most a countable number of different suffices. So, is at most a union of uncountable soft subset, that is, is at most a countable soft subset of . Consequently, if is a soft uncountable soft subset, then it must possess a point of condensation.

Theorem 36. be such that it is second countable. If is uncountable subset of then the soft subset , consisting of all those points of which are not points of condensation of is at most countable.

Proof. Since is second countable space, be a countable open base of . Let be a soft subsequence of the soft sequence , consisting of all those soft sets for which is at most soft countable. Then, is at most countable. We shall establish that . there is not a point of condensation of ; hence, there exists a soft nbhd of , such that is at most countable. Also, there exists a soft set belonging to the soft sequence satisfying Then, is at most countable, and so must be one of the soft sets and therefore . Again, since it follows . Next, let , then belongs to some . Now, as is a soft nbd of and is at most countable. cannot be a point of condensation of . Also, as It follows that Thus, and since each is at most countable. It follows that is also at most countable.☐

3. Main More Results

In this section, 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 and related results, and neutrosophic sequentially compactness are discussed.

Theorem 37. Let be such that it is Hausdorff space and be any . Let :be a soft function such that it is soft monotone and continuous. Then, is also of characteristics of Hausdorffness.

Proof. Suppose such that either . Since is soft monotone. Let us suppose the monotonically increasing case. So, implies that , respectively. Suppose such that , so , respectively, such that Since, is Hausdorff space so there exists mutually disjoint open sets and and. We claim that . Otherwise, . Suppose is soft such that such that Since, is soft one-one implies that implies that . This is a contradiction because. Therefore, Finally, . Given a pair of points . We are able to find out mutually exclusive open sets . This proves that is Husdorff space.☐

Theorem 38. Let be and be another which satisfies one more condition of Hausdorffness. Let a soft function it is soft monotone and continuous. Then, is also of characteristics of Hausdorfness.

Proof. Suppose such that either . Let us suppose the monotonically increasing case. So, implies that , respectively. Suppose such that . So, , respectively, such that such that and . Since but is Hausdorff space. So according to definition or . There definitely exist open sets and such that and and these two open sets are disjoint. Since and are open in Now, and , implies that . We see that it has been shown for every pair of distinct points of , there suppose disjoint open sets, namely, and belong to such that and . Accordingly, is Hausdorff space.

Theorem 39. Let be and be another . Let : be a soft mapping such that it is continuous mapping. Let is Hausdorff space, then it is guaranteed that is a closed subset of

Proof. Given that be and be another . Let : be a soft mapping such that it is continuous mapping. is Hausdorff space. Then, we will prove that is a closed subset of . Equivalently, we will prove that is open subset of Let . Then, or accordingly. Since, is Hausdorff space. Certainly, are points of ; there exists open sets such that and provided . Since is soft continuous, and are open sets in supposing and , respectively, and so is basic open set in containing Since , it is clear by the definition of that is , that is . Hence, implies that is closed.

Theorem 40. Let be and be another . Let : be a open mapping such that it is onto. If the soft set is closed in , then will behave as Hausdorff space.

Proof. Suppose be two points of such that either or Then, or that is or . Since, or is soft in , then there exists open sets and in such that or. Then, since is open, ) and are open sets in containing and , respectively, and ) otherwise or . It follows that is Hausdorff space.

Theorem 41. Let be a second countable space then it is guaranteed that every family of nonempty disjoint open subsets of a second countable space is countable.

Proof. Given that be a second countable space.
Then, a countable base . Let be a family of nonvacuous mutually exclusive open subsets of Then, for each of in , there exists a soft in such a way that . Let us attach with , the smallest positive inter such that . Since the candidates of are mutually exclusive because of this behaviour distinct candidates will be associated with distinct positive integers. Now, if we put the elements of in order so that the order is increasing relative to the positive integers associated with them, we obtain a sequence of candidates of . This verifies that is countable.☐

Theorem 42. Let be a second countable space and let be uncountable subset of . Then, at least one point of is a soft limit point of.

Proof. Let
Let, if possible, no point of be a soft limit point of . Then, for each , there exists open set such that and . Since is a soft base such that . Therefore, . Moreover, if and be any two points such that which means either or , then there exists and in such that and . Now, which guarantees that which implies that
which implies . Thus, there exists a one to one soft correspondence of on to Now, being uncountable, it follows that is uncountable. But, this is purely a contradiction, since being a subfamily of the countable collection This contradiction is taking birth that on point of is a soft limit point of so at least one point of is a soft limit point of ☐

Theorem 43. Let such that is is countably compact then this space has the characteristics of Bolzano Weierstrass Property

Proof. Let be a countably compact space and suppose, if possible, that it negates the Bolzano Weierstrass Property (BWP). Then, there must exists an infinite set supposing no soft limit point. Further, suppose be soft countability infinite soft subset that is . Then, this guarantees has no soft limit point. It follows that is closed set. Also for each is not a soft limit point of Hence, there exists open set such that and . Then, the collection is countable open cover of This soft cover has no finite subcover. For this, we exhaust a single it would not be a soft cover of since then would be covered. Result in is not countably compact.

Theorem 44. Let and be two and suppose be a continuous function such that is continuous function and let supposes the then safely has the

Proof. Suppose be an infinite subset of , so that contains an enumerable set then there exists enumerable set . has implies that every infinite soft subset of supposes soft accumulation point belonging to this implies that has a soft neutrosophic limit point, say, implies that the limit of the soft sequence is . is soft continuous implies thatit is soft continuous. Furthermore, implies that limit of a soft sequence is implies that limit of a soft sequence . Finally, we have shown that there exists an infinite soft subset of containing a limit point This guarantees that has ☐

Theorem 45. Let be a and be subspace of . The necessary and sufficient condition for to be compact relative to is that is compact relative to

Proof. First, we prove that relative to
Let that is be open cover of , then such that implies that there exists such that but . So that . This guarantees that is a open cover of which is known to be compact relative , and hence, the soft cover must be freezable to a finite soft cub cover, say,. Then, is a open cover of . Stepping from an arbitrary open cover of we are able to show that the cover is freezable to a finite soft subcover of meaning there by is compact. The condition is sufficient: suppose be soft subspace of and also is compact. We have to prove that is compact Let be soft open cover of , so that from which . On taking we get . Now from it is clear that is open soft cover of which is known to be compact; hence, this soft cover must be reducible to a finite soft subcover say, . or This proves that is a finite soft subcover to the soft cover . Starting from an arbitrary open soft cover of , we are able to show that this soft neutrosophic open cover is freezable to a finite soft subcover, showing there by is compact

Theorem 46. Let and let be a sequence in such that it converges to a point , then the soft set consisting of the points and is soft compact.

Proof. Given and let be a sequence in such that it converges to a point that is Let . Let be open covering of so that implies that such that According to the definition of soft convergence, implies thatthere exists and . Evidently, contains the points Look carefully at the points and train them in a way as,
generating a finite soft set. Let Then, For this . Hence, there exists such that . Evidently, . This shows that is open cover of Thus, an arbitrary NS open cover of is reducible to a finite cub-cover , it follows that is soft compact.☐

Theorem 47. If such that it has the characteristics of sequentially compactness. Then is safely countably compact.

Proof. Let and let be finite soft subset of Let be a soft sequence of soft points of . Then, being finite, at least one of the elements. In , say must be duplicated an infinite number of times in the sequence. Hence, is a soft subsequence of such that it is soft constant sequence and repeatedly constructed by a single soft number and we know that a soft constant sequence converges on its self. So it converges to which belongs to Hence, is soft sequentially compact.

Theorem 48. Let and be another. Let be a soft continuous mapping of a soft neutrosophic sequentially compact space into . Then, is sequentially compact.

Proof. Given and be another. Let be a soft continuous mapping of a sequentially compact space into . Then, we have to prove sequentially. For this, we proceed. Let be a soft sequence of points in . Then, for each , there exists such that Thus, we obtain a soft sequence in . But being soft sequentially compact, there is a subsequence of such that . So, by continuity of implies that . Thus, is a soft subsequence of converges to in . Hence, is sequentially compact.☐

Theorem 49. Let be a space and such that or . If is a local base at , then there exists at least one member of which does not suppose .

Proof. Since be a space and or open sets and such that but and but . Since, is local base at ; there exists . Since and , so Thus, such that

Theorem 50. Let and suppose be two continuous functions on a in to a which is Hausdorff. Then, soft set is closed of .

Proof. Let if is a set of function. If it is clearly open, and therefore, is closed, which is nothing is proved in this case. Let us consider the case when And let Then, does not belong . Result in Now, being Hausdorff space so there exists open sets and of and , respectively, such that and such that these sets such that the possibility of one rules out the possibility of other. By soft continuity of , as well as is open nhd of , and therefore, so is is contained in for, and because and are mutually exclusive. This implies that does not belong to Therefore, This shows that is nhd of each of its points. So, open, and hence, is closed.

Theorem 51. Let such that it is Hausdorff space and let be a soft continuous function of into itself. Then, the set of fixed points under is a closed set.

Proof. Let . If , then is open, and therefore, closed. So, let and let . Then, does not belong to , and therefore, . Now, and being two distinct points of the Hausdorff space , so there exists open sets and such that and are disjoint. Also, by the continuity of , is open set containing . We pretend that . Since . As implies that does not belong to . Therefore, . Thus, is the nhd of each of its points. So, is open, and hence, is closed.☐

4. Conclusion

Crisp topology is such an important branch of mathematics which is used as applied mathematics as well as pure mathematics. Soft topology is an extension of crisp topology. It actually discusses the behaviour of the subsets of the crisp set with the help of parameters. Fuzzy soft topology only discusses the membership value, and it has nothing to do with the nonmembership value. This is the drawback of fuzzy soft topology. The intuitionistic fuzzy soft topology is an extension of fuzzy soft topology. It addresses both the degree of membership as well as the degree of nonmembership. Intuitionistic fuzzy soft topology is failed to address the indeterminacy case. This deficiency is filled by neutrosophic soft topology. This neutrosophic soft topology addresses acceptance, rejection, and also indeterminacy case. Characterization of neutrosophic soft points, neutrosophic soft separation axioms, countability theorems, and countable space can be Hausdorff space under the restriction of neutrosophic soft sequence which is convergent, cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological characteristics of different spaces, neutrosophic soft continuity, product of different neutrosophic soft spaces, and neutrosophic soft countably compact has the characteristics of Bolzano Weierstrass Property are studied. BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence and its linking with neutrosophic soft compact space, sequentially compactness are addressed.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflict of interest.

Acknowledgments

The authors acknowledge with thanks the support of the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. This research work was partially supported by Chiang Mai University.