An Absolutely New Attempt to Vague Soft Bitopological Spaces Basis at Newly Defined Operations regarding Vague Soft Points

Abstract

In this study, new operations of union, intersection, and complement are defined with the help of vague soft sets in a new way that is in both true and false statements, union is defined with maximum, and intersection is defined with minimum. On the basis of these operations, vague soft topology is defined. Pairwise vague soft open sets and pairwise vague soft closed sets are defined in vague soft bitopological structures (VSBTS). Moreover, generalized vague soft open sets are introduced in VSBTS concerning soft points of the space. On the basis of generalized vague soft open sets, separation axioms are also introduced. In continuation, these separations axioms are engaged with other important results in VSBTS.

1. Introduction

Fuzzy set theory [1] is the most importantly effective way to deal with vagueness and incomplete data, and it is being developed and used in various fields of science. Fuzzy set theory is the most attractive approach to address the uncertain situation. It is an entirely different approach in comparison to probability theory. It may be used in such situations where probability theory is getting failed. However, it has a shortcoming, i.e., it only addresses the membership value and is at a loss to address the nonmembership value. Atanassov [2] bridged the gap in fuzzy set theory with a new way and introduced the concept of intuitionistic fuzzy set theory. This approach supposes membership value and nonmembership value.

Molodtsov [3] shaked the pillars of set theory and installed the concept of soft set theory (SS) to address the uncertainty in a meaningful way. Soft set theory (SST) has a big hand as an application in many fields like function smoothness, Riemann integration, measurement theory, and game theory [4]. The concept of vague set theory which is generalization of fuzzy set theory was suggested by Gau and Buehrer [5]. A vague set is defined as truth-membership function and false membership function.

Wei et al. [6] combined the soft set and vague set structures and leaked out the concept of vague to soft set theory. Huang et al. [7] deeply studied [6] and pointed out some incorrect results. They verified the incorrect result with examples and gave some more new definitions. Wang and Li [8] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. The authors discussed the basic concepts related to vague soft topological and studied the results in vague soft topology. Wang and Li [8] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. They authors discussed the basic concepts related to vague soft topological and studied the results in vague soft topology. For better understanding, the authors provided number of examples.

Mukherjee and Das [9] studied the concept of neutrosophic bipolar vague soft sets and some of its operations. It is the combination of neutrosophic bipolar vague sets and soft sets. Furthermore, the authors developed a decision-making method based on neutrosophic bipolar vague soft set. A numerical example has been shown. Some new operations on neutrosophic bipolar vague soft set have also been designed.

The organization of the study is as follows. We first attempted the fundamentals definitions of vague soft set including complement, subsets, equal sets, null set, absolute set, union, and intersection in a new approach. On the basis of these concepts, vague soft topological structure is defined that are useful for subsequent discussions. Al-Quran and Hassan [10] extended the notion of classical soft set to neutrosophic vague soft sets by applying the theory of soft sets to neutrosophic vague soft sets to make more effective results. They defined new operations. Finally, they applied the theory to decision-making problems. An ample of examples is provided to make the results more effective in this direction.

In our study, we worked with the new operations which are entirely different from references [5, 6]. Then, unlike references [7, 8], vague soft topological structure is reconstructed. In Section 2, some basic operators are introduced in a new way. In Section 3, vague soft bitopology (VSBT) is addressed with ample of examples. Some results, union, and intersection are also studied (VSBT). In Section 4, some main results are addressed. In Section 5, main more results are addressed. In Section 6, some concluding remarks and future work are addressed.

2. Preliminaries

In this section, some fundamental concepts are addressed.

Definition 1. Let be key set (KS) and set of parameters. Let signifies power set of all vague sets on . Then, a vague soft set over is a set defined by a set-valued function representing a mapping , where is called the approximate function of the vague soft set . It can be written as a set of ordered pairs:

Definition 2. Let be a vague soft set over key set The complement of is signified as and is defined as follows: . It is clear that

Definition 3. and be two vague soft subsets over key set . is supposed to be vague soft subset of if , , . It is signified as is said to be equal to if is a vague soft subset of and is a vague soft subset of . It is symbolized as .

Definition 4. Let be two vague soft subsets over key set , so that . Then, their union is signified as and is defined as , where

Definition 5. If , be two vague soft subsets over key set , such that . Then, their intersection is signified as and is defined as , where

Definition 6. If be a vague soft set over key set, is said to be a null vague soft set (NVSS). If .
It is signified as .

Definition 7. If be a vague soft over key set, is an absolute vague soft set if , and .

Definition 8. If VSS be the family of all VS soft sets and , then is said to be a VS soft topology on if belong to , the union of any number of VS soft sets in belong to , and the intersection of a finite number of vague soft sets in belong to . Then, is said to be a vague soft topology over .

Definition 9. Let be the family of all vague soft sets over key set . Then, is called a vague soft point, for every ∈, and is defined as follows:

Definition 10. Let be over KS . if

Definition 11. be a vague soft topological space over and be a vague soft over . Vague soft set in is called a vague soft nbhd of the vague soft point , if there exists a NS open set , such that .

3. Vague Soft Bitopological Structure

In this part, the concept of vague soft bitopological space is defined. Furthermore, new types of open and closed sets have been introduced in the same space. Example of vague soft bitopological space is given. Intersection of vague soft bitopological space is addressed. Union of two vague soft bitopological spaces need not be vague soft topological space. This situation is established with example. Pairwise vague soft open and soft closed sets are addressed in vague soft bitopological space. Some theorems related to this concept are also addressed.

Definition 12. If and are two VSTS, then is called VSBTS. Let be vague soft bitopological space. A vague soft set is said to be open in if there exists a vague soft open set in and vague soft open set in , such that . The vague soft set is said to be close in if its complement is open in .

Example 1. Let , , , and , where , and being vague soft sets are as follows:Then, , , , , and , , , , .
Therefore, , are on , so is VSBTS.

Theorem 1. Let be a VSBTS. Then, is on .

Proof. (1) and (3) are obvious. For (2), we proceed as follows: let . Then, and . Since , are vague soft topologies on , then and . Therefore, .

Remark 1. Let be VSBTS; then, need not be VSBTS on .

Example 2. Let , , and , where , and being vague soft sets are as follows:Here, is not a vague soft topology on . Since, .

Definition 13. Let be VSBTS. Then, a vague soft setis called as a pairwise vague soft open set if there exist a vague soft open set in and a vague soft open set in , such that for all ,

Definition 14. Let be VSBTS. Then, a vague soft setis called as PVSOS if there exist a vague soft open set in and a vague soft open set in , such that for all ,The set of all pairwise vague open sets in is denoted by PVSO .

Definition 15. Let be VSBTS. Then, a vague soft setis called as a pairwise vague soft closed set (PVSC) if is . It is clear that is a PVSC set if there exist in and VSOC in , such that for all ,The set of all in is denoted by .

Theorem 2. be . In this case,(1)(2)If , then (3)If , then

Proof. (1)Since , , then and are (2)Since , there exist , , such that for all . Then, As , are on , and  Therefore, (3)Since , there exist and , such that for all . Then,Then, as , and .

Definition 16. Let be . closure of , denoted by , is the intersection of all containing , i.e.,It is clear that is the smallest containing .

Theorem 3. Let and . Then,(1) and (2)(3) is a PVSCS if (4) if (5)(6), i.e., is PVSCS

Proof. It is obvious.

Theorem 4. Let be , . Then, if and only if for all , where is any containing , and is the family of all containing , .

Proof. Let , and suppose that there exists , such that . Then, . Thus, , which implies , a contradiction.
Conversely, ; then, . Therefore, by hypothesis, , a contradiction.

4. Main Results

In this section, some new definitions are introduced in vague soft bitopological spaces. All these definitions are the backbone for the upcoming study. Among these definitions, definition (3) became source of motivation for our results which are discussed in this section and in the next section. All the results discussed in this section are discussed with respect to soft points of the spaces.

Definition 17. Let be over be a set over . Then, is(1)Vague soft semiopen if (2)Vague soft preopen (3)Vague and vague soft close if .

Definition 18. Let be over , are VS points If there exist open sets and , such that . Then, is called a space.

Definition 19. Let be over , and are points. If there exist -open sets , such that and . Then, is called a space.

Definition 20. Let be over are points. If open set , , such that , , and . Then, is called a space.

Example 3. Let , the set of conditions by . Let us consider , , , and be points. Then, the family , , is Also, . Thus, be . Also, is structure.

Theorem 5. Let be . Then, be a structure if and only if each point is -closed.

Proof. Let be over and be an arbitrary point. We establish is a soft -open set. Let . Then, either or or . This means that are two distinct points. Thus, or or or or or or or . Since be a structure, there exists a -open set , so that and . Since, . So, . Thus, is a -open set, i.e., is a -closed set. Suppose that each point is -closed. Then, is a -open set. Let . Thus, , and . So, be a - space.

Theorem 6. Let be over the father set . Then, is space if and only if for distinct points , , there exists a -open set , containing but not , such that .

Proof. Let be two points in space. Then, there exists disjoint open sets , such that . Since implies . Next, suppose that , there exists a -open set containing but not , such that , that is, and are mutually exclusive -open sets supposing and in turn.

Theorem 7. Let be Then, is space if every point , if there exists a open set , such that . Then, is a space.

Proof. Suppose . Since, is space. and are the -closed sets in . Then, . Thus, there exists a -open set , such that . So, we have and , that is, is a soft space.

Definition 21. Let be . be a -closed set, and . If there exists -open sets and , such that , and , then is called a -regular space. is said to be space, if it is both a regular and space.

Theorem 8. Let be is soft space if and only if for every , that is, (such that .

Proof. Let is space and . Since is space for the VS point , and is a closed set and , such that , and . Then, we have . Since is a closed set, . Conversely, let be a closed set. , and from the condition of the theorem, we have . Thus, . So, is space.

5. More Main Results

In this section, more main results are addressed. These results are discussed between two structures. The behavior of one structure can safely be transferred from one space into another space through soft functions by putting some conditions on the soft function. Similarly, through soft function, the engagement between structures and soft closed sets can be studied. The engagement is between second countable and soft limit points. The study of B. V. property, soft vague compactness, sequentially compactness, and countably compactness results are addressed with respect to soft points of the spaces.

Theorem 9. Let be , such that it is Hausdorff space and be another . Let : which is a soft function, such that it is soft monotone and continuous. Then, is also the characteristics of Hausdorffness.

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

Theorem 10. Let be and be another which satisfies one more condition of Hausdorffness. Let : be a soft function, , it is soft monotone and continuous. Then, is also characteristics of Hausdorffness.

Proof. , such that either . Let implies that or , respectively. Suppose , . So, , respectively, such that , such that and . Since , but is Hausdorff space. So according to definition, or . There definitely exists -open sets and , such that and , and these two -open sets are disjoint. Since, and are open in . Now, and , imply that . , . Accordingly, is Hausdorff space.

Theorem 11. Let be and be another . Let : be a soft mapping. Let be 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 a -open subset of . Let . Then, or accordingly. Since, is V-Hausdorff space. Certainly, are points of , and there exists -open sets , such that , , provided . Since is soft continuous, and are -open sets in , supposing and , respectively, and so, is a basic -open set in containing . Since , it is clear by the definition of , that is, , that is, . Hence, implies that is closed.

Theorem 12. Let be and be another . Let  :  be 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 13. Let be second countable space, and let be a VS 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 a -open set , such that and . Since is soft base , such that . Therefore, . Moreover, if and be any two points, such that which means either or , then there exist and in , such that and . Now, , which guarantees that , which implies that , which implies . Thus, -to-one soft correspondence of on to . Now, being uncountable, it follows that is uncountable.

Theorem 14. Let and be two and suppose be a continuous function, such that is a continuous function, and let suppose the Then, safely, has the .

Proof. Suppose be an infinite subset of , so that contains an enumerable set ; then, there exists a enumerable set s.t., . has which implies that every infinite soft subset of supposes soft accumulation point belonging to ; this implies that has soft vague limit point, say, implies that the limit of soft sequence is . f is soft continuous implying that it is soft continuous. Furthermore, implies that limit of a soft sequence is , implying 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 15. 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. 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 that there 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 VS open cover of is reducible to a finite subcover , and it follows that is soft compact.

Theorem 16. If , it has the characteristics of NS 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 in 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 single soft number , and we know that a soft constant sequence converges on itself. So, it converges to , which belongs to . Hence, is soft sequentially compact.

Theorem 17. Let and be another . Let be a soft continuous mapping of a soft vague 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 VS sequentially. For this, we proceed as 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 that converges to in . Hence, is sequentially compact.

Theorem 18. Let , and suppose be two continuous function on BTS into , which is Hausdorff. Then, soft set is closed of .

Proof. Let is a VS set of function. If , clearly open, and therefore, is closed, that is, nothing is proved in this case. Let us consider the case when . Let . Then, does not belong to . Result in . Now, be Hausdorff space, so there exists open sets , of and , respectively, such that and , and these sets are mutually disjoint. By soft continuity of , as well as is open neighborhood of , and therefore, so is contained in for and because and are mutually exclusive. This implies that does not belong to . Therefore, . This shows that is neighborhood of each of its points. So, open, and hence, is closed.

Theorem 19. Let , such that it is Hausdorff space, and let be soft continuous function of into itself. Then, the set of fixed points under is a

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

6. Conclusion

Vague soft topology is an extension of fuzzy soft topology. Fuzzy soft topology is directly an extension of the crisp topology which has membership functional candidates bound by length one. However, it has a shortcoming, i.e., it only addresses a membership value and is unable to address the nonmembership value. Vague soft topology is dominant over fuzzy soft topology because it supposes all the two information that is true and false at the same time. Vague soft topology has narrowed domain as compared to vague soft bitopology. Some problems are very hard to discuss in vague soft topology. So, need of vague soft bitopology is felt. In our work, we regenerated some structures in vague soft bitopological spaces with new definition, that is, open sets concerning soft points. In future, we would try to see the validity of the given structures in vague soft tritopological structures under soft points of the space under the application of some more generalized vague soft open sets.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors read and approved the final manuscript.