Abstract
Topology is studying the objects which are considered to be equal if they may also be continually deformed through other shapes as bending and twisting without tearing or glueing them. Topology is similar in geometrical structures and quantitatively equivalent. Nanotopology is the study of set. The main goal of this article is to propose the idea of generalized closed sets in Pythagorean nanotopological spaces. In addition, the concept of semigeneralized closed sets is also defined, and their properties are investigated. An application to MADM using Pythagorean nanotopology has been proposed and illustrated using a numerical example.
1. Introduction
Topology is a discipline of mathematics in which two objects are regarded equal if they may be continously deformed into one another through motions in space such as bending, twisting, stretching, and shrinking without preventing tearing apart or glueing together sections. The qualities that stay constant by such continuous deformations are the core issues of interest in topology. While topology is similar to geometry, it varies in that geometrically equal things generally share numerically measured properties such as lengths or angles, whereas topologically analogous objects are qualitatively equivalent. General topology is the branch of topology that deals with the fundamental set-theoretic notions and constructs used in topology in mathematics. Most other fields of topology, such as differential topology, geometric topology, and algebraic topology, are built on it. Continuity, compactness, and connectedness are the three fundamental principles in point-set topology. Intuitively, continuous functions transport nearby points to nearby points. Compact sets can be covered by an infinite number of small sets. Connected sets are those that cannot be separated into two separate pieces.
Fuzzy set theory [1] plays a vital role in dealing with incomplete data and vagueness, and it is applied in a wide range of disciplines. Fuzzy set is an extension of the usual set holding elements with its membership grade in the interval [0,1]. Along with some conditions, there has been an advancement in fuzzy set (intuitionistic fuzzy set (IFS)) in the view of other human thinking options [2]. To each element in the IFS, it has membership and nonmembership grades which satisfy the condition that the sum of both the grades is lesser than or equal to 1. The Pythagorean fuzzy subset (PFS), an advancement of the fuzzy subset with various applications, was presented by Yager [3, 4]. PFS can be used in any situation where IFS is not appropriate.
Fuzziness was improved from intuitionistic and further extended to neutrosophic sets. Smarandache [5] presented neutrosophic sets, a crucial mathematical concept for dealing with indeterminate, and inconsistent data. The set that assigns truth, indeterminacy, and false membership grades for elements that assume values within the interval characterizes a neutrosophic set (NS). Wang et al. instituted the generalization of intuitionistic sets and a sub of NS, single-valued NS in [6] which has elements with three membership grades holding the values in interval [0,1].
Chang defined fuzzy topology in [7] as a collection of fuzzy sets that satisfy the axioms of topological spaces. In topology, the fuzzy set theoretical concepts were applied and various notions of topological space were introduced as convergence and compactness [8–10]. Following this, intuitionistic topological spaces were developed into ideas as separation axioms, connectedness, and categorical property [11–15].
Using an equivalence relation in a subset of universe in terms of boundary region and approximations, nanotopological space was introduced. Subsequently, functions on nanotopological spaces, namely, nanocontinuous functions and their characterizations in forms of nanoclosed sets, closure and interior were derived [16]. Weak forms of open sets as nanoaplha open, semiopen, and preopen sets with various form of nano--open and semiopen sets corresponding to various case of approximations were derived in [17]. In [18, 19], the concept of nanocompactness and connectedness, generalized closed sets were developed with their properties. The nanosemipreneighbourhoods, semipreinterior, semiprefrontier, semipreexterior, nanogeneralized preregular closed sets were defined, and relations between the existing sets have been examined in [7, 8].
The notion of intuitionistic fuzzy nanotopological space was introduced, and the weak forms of intuitionistic fuzzy nanoopen sets and properties of intuitionistic fuzzy nanocontinuous functions are investigated in [20]. Intuitionistic fuzzy nanogeneralized continuous mappings and closed sets were defined, and their properties were examined in [21, 22]. Thivagar et al. presented the idea of nanotopology neutrosophic units in [23]. [24] introduced the Pythagorean fuzzy topological space by following Chang. By making a fusion of the concepts Pythagorean topological space and nanotopological space, Pythagorean nanotopological spaces (PNTSs) were developed in [25–28].
Multicriteria decision-making is a branch of operation research. Decision-making often involves vagueness which can be effectively handled by fuzzy sets and fuzzy decision-making techniques. In recent years, a great deal of research has been carried out on the theoretical and application aspects of MCDM and fuzzy MCDM. The algorithms of the popular MCDM processes are AHP and TOPSIS. Subsequently, fuzzy MCDM techniques are introduced, and their applications in different disciplines are more effective nowadays. MDCM in general is as follows: problem formulation, identification of the requirements, goal setting, identification of various alternatives, development of criteria, and identification and application of decision-making technique. Various mathematical techniques can be used for this process, and the choice of techniques is made based on the nature of the problem and the level of complexity assigned to the decision-making process. All methods have their own pros and cons. According to a recent literature review by [29], there were more than hundreds of research articles published in the last two decades showing the application of MCDM. The development of the fuzzy decision-making and its tremendous growth is discussed in detail in the review by Mardani et al. [30]. As the fuzzy set has been developed into many fuzzy sets, the MCDM has also been evolved around those sets and transformed into a usable tool in the application for different disciplines. Recently, the MCDM has been developed and used in applications as in [31–35]. Our motivation for the work is that this is still a developing area in fuzzy mathematics, and we want to produce more theoretical concepts and show the application of the work in some real-life situations by combining it with wide-area decision-making. There are many existing models which are still developing in this particular area but when we deal with more fuzzified data, this method is more useful without reducing the constraint when compared to the other concepts. The proposed concepts and model have the more fuzzified values as information but still hold the same condition as the other models, which has a great advantage in dealing with the more vague details of the problem.
The following is how the article is organized: In Section 2, we define generalized closed sets of PNTS along with its characterizations. Sections 3 discusses the generalized semiclosed sets of PNTS. In Section 4, we present an MADM algorithm by using Pythagorean nanotopology, illustrate with the help of numerical example, and conclude in Section 5.
2. Pythagorean Nanogeneralized Closed Sets
has been defined in [28], the weak forms of open sets of have been defined, and their properties were investigated in [26, 28]. In this section, as an extension of these ideas of , the generalized closed sets have been developed and various characterizations of these sets have been examined. Throughout this paper, Pythagorean nano is denoted by .
Definition 1 (see [36]). Let the Universe be and equivalence relation on be , and if where , holds the following axioms:
(1)(2)If for, then (3)If for , then Then, is termed as topology () on w.r.t whereas , , and where and
We call as Pythagorean nanotopological spaces (). The elements of are called open () sets.
Definition 2. A subset of is generalized closed () if whenever , is in
Theorem 3. is in a iff has no nonvoid set.
Proof. Let be a and be subset of Then, , and is . Since is , Therefore, and . That is, ; thus, cannot have any nonempty set.
Conversely, let has no nonvoid set. be a such that . Then, is a subset of , since , as Therefore, Since does not have any nonempty set. Hence, Therefore, is in ☐
Theorem 4. Every set is .
Proof. If is , then Therefore, for every set such that and hence is .☐
Theorem 5. A set is in if and only if is .
Proof. If is in , then and hence which is . Conversely, is , and Therefore, , and hence, is .☐
Theorem 6. If and are , then is .
Proof. Let be in such that Then, and since is and is a set having
Similarly, is .☐
Theorem 7. Let be a subset of and be a set to Then, is to
Proof. Let be a set in such that Then, is a to containing Since is to , where is the closure of in Then, and Also, is . Since is in Therefore, , since Therefore, Thus, for every set in containing , Therefore, is in ☐
Now, we prove the intersection of a and a is .
Corollary 8. If is and is in then is .
Proof. Since is to , it is to Since where is in and is to , is in by the above theorem.☐
Theorem 9. If is and , then is .
Proof. Since is , has no nonempty subset. Since , also does not have any nonempty set. Therefore, is .☐
Theorem 10. Let and be two spaces and and be in Then, is in
Proof. Let be in such that Then, where is in and Therefore, That is, is containing Since is in Therefore, That is, , for every in such that Therefore, is in ☐
Theorem 11. Every subset of is iff
Proof. Let and Let be in such that Then, , since , , , and are the only sets which are as well as in , when Thus, whenever is and is , if ☐
Conversely, assume that every subset of is . Let Then, is . Since is and , ; hence, is . Thus, whenever is in , and is in The closure of each set in is . is extremally disconnected, and hence,
Definition 12. A set in a is generalized open () if is .
Theorem 13. Every set is .
Proof. We know that every is . Thus, if we take complement, we get every is . But the converse need not to be true. That is, every need not be .☐
Theorem 14. is iff whenever is and
Proof. Let be a in Then, is . Therefore, whenever is and Let be and Then, , and is . Therefore, That is, Hence,
Conversely, let for every set such that , and let be such that Then, where is . Therefore, Therefore, Therefore, That is, Therefore, is , and hence, is in ☐
Definition 15. If and are subset of , then and are said to be separated (), if and
Theorem 16. If and are and , then is .
Proof. Let be in such that Since and are , Therefore, no element of belongs to Thus, no element of belongs to Hence, every element of belongs to , since That is, Thus, is subset of Since is ,
Similarly, Since . Thus, , whenever is and ☐
Therefore, is .
Theorem 17. If and are in , then is .
Proof. and are and hence is and hence is .☐
Theorem 18. If and are sets such that and are , then is .
Proof. and are and , and hence, is . Therefore, is .☐
Theorem 19. is iff where is and .
Proof. Let be . Let be such that Then, Since is , cannot have any nonempty set. But is subset of Therefore, That is,
Conversely, assume that whenever is and , then . Let be such that . Then, which is . Therefore, That is, , since every , belongs to . Thus, whenever is and Therefore, is .☐
Theorem 20. If and if is , then is also .
Proof. where is , and hence, is . Therefore, is .☐
Theorem 21. is if and only if is .
Proof. Let be . Let be a such that Then, , since cannot have any nonempty closed set. Therefore, , and hence, is .
Conversely, if is and is such that , then where is . Since is , Therefore, , and hence, Thus, whenever is and , is .☐
Theorem 22. In a (, if , then any set such that is the only set in
Proof. When , and are the only sets and hence for any subset of , is the only set holding it. Therefore, for every set having Thus, every subset of is PNg-C, if . When , the sets in are , , and . If , then sets having are and . Also, And . Therefore, is not . If , then is the only set holding and hence for every set . Therefore, is . Thus, only those sets such that are , if .☐
Theorem 23. If and in a , then those sets for which are the only sets.
Proof. If , then and are the set containing . ; hence, . Thus, when . Therefore, is not . But, if , then is the only set that contains and hence whenever is and Therefore, is . Thus, only those subsets of such that are in , if and ☐
Theorem 24. If and , then every subset of is .
Theorem 25. If and , , and only those subsets of such that are in
Proof. . If , then sets containing are , , and . But , since . Therefore, is not . If , then , , and are the sets containing and Therefore, is not . If , neither a subset of is a set containing for which . Therefore, is not . If , then is the only set containing and hence for every . Therefore, is . Thus, only those for which are in .☐
3. Pythagorean Nanosemigeneralized Closed Sets
As we have defined in the last section, we have extended the concept of generalized closed sets to semigeneralized closed sets and investigated their properties.
Definition 26. A subset of is said to be semigeneralized closed (), if whenever is and The set is named as semigeneralized open () if is .
Definition 27. If , then the semigeneralized closure represented by is defined as the smallest set having
The semigeneralized interior of , symbolized by , is defined as the largest set in
Remark 28. For subsets and of a , (1)(2)
Theorem 29. Every set is .
Theorem 30. A set is in iff has no nonempty, set.
Proof. Let be a and be a subset of Then, , and is . That is, . Therefore, . Thus, is , and Since is , . That is, . Thus, Therefore,
Conversely, let have no nonempty, set. Let be in such that . If , then . And , since . Thus, is a nonvoid subset of , which is contradiction. Therefore, whenever is and . That is, is in ☐
Theorem 31. Let be . Then, is iff is .
Proof. Let be . If if , and hence which is . Conversely, let be . Then, is . Then, does not contain any nonempty, set. Therefore, . That is, . Therefore, is .
Now, we derive the forms of semigeneralized closed sets for various cases of approximations.☐
Theorem 32. If in a , then any and where are the only sets in
Proof. When , Also, and any are the only sets in If , then and the sets containing are those sets for which . Thus, , not for every such that .
Therefore, is not . If , then , since any subset of is in Thus, whenever is and Therefore, is . If and , and the sets containing are and . Therefore, . Therefore, any and are not . If , then whenever is and , since is the only set containing . Therefore, if , then is . When has at least one element of and exactly one element of where is not a singleton set, But union of that element and is a set containing and union with that element. Therefore, is not . Thus, the only sets in are subsets of and any ☐
Theorem 33. If and , then the only sets in are subsets of and any
Proof. If and , then . Also, and those sets for which are the only sets in Therefore, the sets for which are the only sets in If , then But is a set containing for which Therefore, is not . If and , then But, for , which is set containing itself, Therefore, is not . If , then and hence for every set such that , Therefore, is . If , then is the only set holding and hence whenever is and Therefore, is . If has one element of and at least one element of , then Since any set having is in ,
and any set having are sets containing But, Therefore, is not in Thus, only subsets of and any are in when and ☐
Theorem 34. If and in a , then every subset of is .
Proof. , , , and are the only sets in which are , , and in If , then and are the only sets containing and Therefore, whenever is and Thus, is . If , then and are the only sets containing and Therefore, is PNsg-C. If or , then is the only set containing and hence is . If contains atleast one element of and at least one element of , then is the only set containing Therefore, is . Thus, every subset of is , if and ☐
Definition 35. Let and be two PNTSs. Then, a function is named as (1) generalized continuous (), if the inverse image of every set in is in (2) semigeneralized continuous (), if the inverse image of every set in is in (3) semigeneralized closed () if the image of every set in is in (4) semigeneralized open () if the image of every set in is in
Theorem 36. Every continuous () function is .
Proof. If is on and if is in , then is in Therefore, is , since any set is and any is PNSgO. Therefore, is .☐
4. Pythagorean Nanotopology in Multiple Attribute Decision-Making
MADM is a method for selecting the best solution with the highest level of satisfaction from a set of alternatives. Multiple attributes are used to represent these types of MADM problems, which occur in most real-time situations. When it comes to dealing with real-life problems, collecting vague details is done with the help of attributes for the particular object and the decision-making technique is applied for the list of objects considered. Many models already exist for the decision-making problems, but the proposed algorithm deals with membership and nonmembership which has more advantage in fuzziness than intuitionistic fuzzy set and fuzzy set theory. Many types of models exist for the different developments of topological spaces, but for the different category of topological spaces, this method is proposed. The proposed algorithm describes how topology influences decision-making.
A new decision-making approach using topology and a methodological approach for selecting the right alternatives is proposed.
4.1. Algorithm
Step 1. Consider the universe and attributes .
Step 2. Make a fuzzy Pythagorean matrix of attributes versus objects.
Step 3. Define on to represent the indiscernibility relation.
Step 4. Build the Pythagorean fuzzy nanotopology
Step 5. Find the score values by using the score function (where means membership, n.m means nonmembership, and is the number of values in the corresponding topology) of each of the entries of Pythagorean fuzzy nanotopological spaces.
Step 6. Arrange the score values of the alternatives in decreasing order and select the maximum as the optimal decision.
The pieces of information for the object are collected for the particular object and formed the table, and after that, using the relation, the PN topology is being framed. Using score function, the optimal values are calculated and the decision is made upon the maximal value.
4.2. Numerical Example
The proposed algorithm helps to find the suitable choice among all the options (set of objects). We choose any random situation for this decision-making process. As in Algorithm, using the method, the problem is solved. Let us consider the decision-making situation where a company in a tourist hotspot desires to select and draw a contract with a hotel for certain years. Let us consider the set of objects as the hotels which were considered to have a contract. That is, where . Consider the criteria for deciding to pick a hotel. The attributes are where stands for criteria clean and tidy, good food, reasonable price, customer driven, and location, respectively.
Step 1. Let be the set of objects and be the set of attributes for the objects.
Step 2. In the matrix of Pythagorean fuzzy relationship between hotels, attributes are developed as in Table 1.
Step 3. The indiscernibility relation for the objects is constructed as
Step 4. Build the for each hotel with respect to the attributes.
Step 5. Computation of Pythagorean fuzzy score functions for the hotels as in algorithm are as follows:
Score values of hotels are
Step 6. Organizing the score values, we get the sequence of the hotels as . Thus, the hotel with maximum value and in the first position is chosen as the optimal decision (i.e., ).
4.3. Comparison Analysis
To check the effectualness of the presented decision-making approach, a comparison analysis is performed with Pythagorean fuzzy decision-making model used in [36]. Though the ranking principle and method are different, the ranking order results are consistent with the result obtained in [36] for the selection of the best alternative. The computation may seem hard, but the calculation is too easy to compute, while when compared to the three-valued sets, this possesses a little lack in the indeterminacy part. When compared to the other sets and models, this plays the upper hand.
5. Conclusion
is a newly defined space by combining the concepts of nanotopology and Pythagorean fuzzy topological spaces. The topological space has been developed, and as an extension, the concepts of the weak open sets, namely, nanoalpha, semiopen sets, have been developed and their characterizations were examined. In this article, the idea of generalized closed sets in Pythagorean nanotopology has been introduced along with its characteristics. The notion of semigeneralized closed sets has also been defined, and their properties were investigated. An application in MADM using has been proposed and illustrated using a numerical example. Further, the proposed concept can be extended to strong open sets in and applied to real-life problems.
Data Availability
No data were used to support this study.
Disclosure
The statements made and views expressed are solely the responsibility of the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.