Abstract
In this paper, we introduce fuzzy nano (resp. δ, δS, P and Z) locally closed set and fuzzy nano (resp. δ, δS, P and Z) extremally disconnected spaces in fuzzy nano topological spaces. Also, we introduce some new spaces called fuzzy nano (resp. δ, δS, P and Z) normal spaces and strongly fuzzy nano (resp. δ, δS, P and Z) normal spaces with the help of fuzzy nano (resp. δ, δS, P and Z)-open sets in fuzzy nano topological space. Numerical data is used to quantify the provided features. Furthermore, using fuzzy nano topological spaces, an algorithm for multiple attribute decision-making (MADM) with an application in medical diagnosis is devised.
1. Introduction
Through his significant theory on fuzzy sets, Zadeh [1] made the first effective attempt in mathematical modeling to contain non-probabilistic uncertainty, i.e. uncertainty that is not caused by randomness of an event. The study of fuzzy calculus plays a vital role in the field of mathematics due to its useful applications in variety of scientific domains including statistics, applied mathematics, dynamics and mathematical biology. Many applications of fuzzy mathematics can be found in engineering, bio-mathematics and basic sciences. A novel technique to solve the fuzzy system of equations has been presented by Mikaeilvand et al. [2]. Also many applications of fuzzy integral equations have been presented by various authors [3, 4]. A fuzzy set is one in which each element of the universe belongs to it, but with a value or degree of belongingness that falls between 0 and 1, and these values are referred to as the membership value of each element in that set. Chang [5] was the first to propose the concept of fuzzy topology later on.
Pawlak [6] introduces Rough set theory in 1992 as a substitute mathematical tool for describing reasoning and deciding how to handle vagueness and uncertainty. This theory uses equivalence relations to approximate sets, and it is used in conjunction with the principal non-statistical techniques to data analysis. Lower and upper approximations are two definite sets that commonly characterise a rough set. The greatest definable set included inside the given collection of objects is the lower approximation, whereas the smallest definable set that contains the provided set is the upper approximation. Rough set concepts are frequently stated in broad terms using topological operations such as interior and closure, which are referred to as approximations.
Lellis Thivagar [7] introduced a new topology called nano topology in 2013, which is an extension of rough set theory. He also created Nano topological spaces, which are defined in terms of approximations and the boundary region of a subset of the universe using an equivalence relation. The Nano open sets are the constituents of a Nano topological space, while the Nano closed sets are their complements. The term “nano” refers to anything extremely small. Nano topology, then, is the study of extremely small surfaces. Nano topology is based on the concepts of approximations and indiscernibility relations. In addition, in [8], nano open sets in nano topological space were investigated.
This paper follows the definition of Lellis Thivagar et al. [9]. Generalizations of (fuzzy nano) open sets are a major topic in (fuzzy nano) topology. One of the important generalizations is a -open sets [10] which was studied in classical topology by El-Magharabi and Mubarki. Later on, many studies which investigated a nano topologies have been done such as nano -open sets [11], nano -open sets [12], -closed sets in double fuzzy topological spaces [13, 14] and -open sets in a fuzzy nano topological spaces by Thangammal et al. [15].
Kuratowski and Sierpinski [16] explored the difference of two closed subsets of a -dimensional Euclidean space in 1921, and the notion of a locally closed subset of a topological space was a key instrument in their work. Ganster and Reilly [17] defined -continuity in a topological space using locally closed sets in 1989.
Multiple attribute decision-making (MADM) is a decision-making process that takes into account the best possible options. Decisions were taken in mediaeval times without taking into account data uncertainties, which could lead to a potential outcome. Inadequate outcomes have real-life consequences. If we deduced the consequence of obtained data without hesitancy, the results would be ambiguous, indeterminate, or incorrect. Without hesitation, I determined the result of the obtained data. MADM had a significant impact on Management, disease diagnosis, economics, and industry are examples of real-world problems. Each decision maker makes hundreds of decisions each time to carry out the key component. It should be a logical assessment of his or her job. MADM is a programme that helps you tackle difficult problems. For this, there are complex problems with a variety of parameters. The problem must be identified in MADM by defining viable alternatives, assessing each alternative against the criteria established by the decision-maker or community of decision-makers, and finally selecting the optimal alternative. To deal with the complications and complexity of MADM problems, a range of useful mathematical methods such as fuzzy sets, neutrosophic sets, and soft sets were developed.
Zafer et al. [18] introduced and developed the MADM method based on rough fuzzy information. Several mathematicians have worked on correlation coefficients, similarity measurements, aggregation operators, topological spaces, and decision-making applications in this area. These structures feature better decision-making solutions and provide distinct formulas for diverse sets. It has a wide range of applications in domains such as medical diagnosis, pattern identification, social sciences, artificial intelligence, business, and multi-attribute decision making. The problems associated with these cases are interesting, and developing a hypothesis for them has prompted many scholars [19–21] to pay attention to them Motivation and objective. No investigation on fuzzy nano locally closed set, fuzzy nano extremally disconnected spaces, fuzzy nano normal spaces and strongly fuzzy nano normal spaces in fuzzy nano topological space has been reported in the fuzzy literature. We present this innovative notion of fuzzy nano topological space and apply it to the MADM issue based on the concepts of fuzzy sets [1], nano topological spaces [7], and neutrosophic nano topological space [9]. The enlarged and hybrid motivation and goal work is described in detail throughout the article. Under certain conditions, we ensure that other FS hybrid systems are special . Our proposed model and techniques are discussed in terms of their robustness, durability, superiority, and simplicity. This is the most prevalent model, and it is used to collect vast amounts of data in AI, engineering, and medical applications. Similar research can simply be duplicated in the future using alternative methodologies and hybrid structures.
The following is how this article is organised: Section 2 is devoted to discussing various fuzzy set theory and fuzzy nano topology definitions and results. In Section 3, we introduce the notion of fuzzy nano locally closed set and establish some of characterizes. The concept of fuzzy nano extremally disconnected spaces is introduced in fuzzy nano topological spaces and also gives some properties and theorems of such concepts in Section 4. In Sections 5 and 6, fuzzy nano normal space and strongly fuzzy nano normal spaces are introduced and proved many theorems. As a numerical example, in Sections 7 8, we devised a method for solving the MADM issue related to Medical Diagnosis utilising . We also discussed the algorithms’ efficiency, advantage, consistency, and validity. In Section 9, the work’s conclusion is fundamentally summarised, and the next field of research is offered.
2. Preliminaries
This part explains the concepts and findings that we need to know in order to comprehend the manuscript.
Definition 1 (see [1]). A function from into the unit interval is called a fuzzy set (briefly, ) in .
Definition 2 (see [1]). If and are any two fuzzy subsets (briefly, ) of a set , then
(i) iff , in . (ii) , if in . (iii) , in . (iv) , in .
Definition 3 (see [1]). The complement of a in , denoted by , is the of defined by , in .
Definition 4 (see [9]). Let be a non-empty set and be an equivalence relation on . Let be a in with the membership function . The fuzzy nano lower (upper) approximations and fuzzy nano boundary of in the approximation denoted by and are respectively defined as follows: (i) (ii) (iii) where . . The collection forms a topology called as fuzzy nano topology and as fuzzy nano topological space (briefly, ). The elements of are called fuzzy nano open (briefly, ) sets. Elements of are called fuzzy nano closed (briefly, ) sets.
3. Fuzzy nano locally closed sets
The idea of fuzzy nano locally closed sets, which represents a class of generalisations of fuzzy nano open sets, is introduced in this section. The main features of fuzzy nano closed sets are established, as well as certain characterizations.
Definition 5. Let be a with respect to where is a fuzzy subset of . Let be a fuzzy subset of . Then fuzzy nano(i)interior of (briefly, ) is represented as .(ii)closure of (briefly, ) is represented as .(iii)regular open (briefly, ) set if .(iv)regular closed (briefly, ) set if .(v) interior of (briefly, ) is represented as .(vi) closure of (briefly, ) is represented as .(vii)-open (briefly, ) set if .(viii)-semi open (briefly, ) set if .(ix)pre open (briefly, FNPo) set if .(x) semi interior of (briefly, ) is represented as .(xi) semi closure of (briefly, ) is represented as .(xii)pre interior of (briefly, FNPint(S)) is represented as .(xiii)pre closure of (briefly, FNPint(S)) is represented as .The complement of an (resp. ) set is called a fuzzy nano (resp. fuzzy nano -semi fuzzy nano pre) closed (briefly, (resp. ) in . Definition 6. Let be a . Then a fuzzy subset in is said to be a fuzzy nano(i)-open (briefly, ) set if ,(ii)-closed (briefly, ) set if .(iii)-interior (resp. closure) of is the union (resp. intersection) of all (resp. ) sets contained in and denoted by (resp. ).All (resp. ) sets of a space will be denoted by (resp. ).
Remark 1. The following diagram shows the relationship between any set in of ’s (resp. ’s).
Definition 7. A function is said to be fuzzy(i)nano (resp. ) continuous (briefly, (resp. )), if set of , the set is (resp. ) set of .(ii)nano irresolute (briefly, ) function, if subset of , the set is subset of .(iii)nano (resp. δ, δS, P and ) open map (briefly, and )) if the image of each set in is and ) in .(iv)nano (res δ, δS, P and ) closed map (briefly, and )) if the image of each set in is and ) in .
Definition 8. Let and be any two fuzzy subsets of a ’s. Then is fuzzy nan δ, δS, P and ) -neighbourhood (briefly, - (resp. , , , a and -)) with if there exists a and ) set with .
Definition 9. Let be a is called fuzzy nano (resp. δ, δS, P and ) locally closed (briefly, (resp, , a and )) set if where is a (resp. , , and ) set and is a (resp. , , a and ) set.
Example 1. Assume and .
Let be a of .Thus .
Then(i)(ii)(iii).(iv).(v).
Proposition 1. Let be a . (i) Every (resp. ) set is set. (ii) Every (resp. ) set is set. (iii) Every (resp. ) set is set. (iv) Every (resp. ) set is set. (v) Every (resp. ) set is set.
Proof. (i) Let be a set in . Then can be written as , where is a set and is a set. Therefore is a set. The rest of the cases are the same.
Proposition 2. Let be a . Every set is (resp. and ) set.
Proof. Let be a set in . Then can be written as , where is set and is set. Since every set is , is the intersection of set and set and hence is set. The rest of the cases are the same.
Remark 2. The converse of the preceding proposition does not have to be true, as the following example demonstrates.
Example 2. In Example 1, . Then is but not .
Theorem 1. Let be a . Then is (resp. ,Every (resp. ) set is ) if and only if (resp. , and therefore for some ) for some (resp. , implies ) set .
Proof. Let be a set. Then , where is set and is set in . Since and so . Olso and implies and therefore for some set . Conversely, assume . Since is set and is set, is set.
The rest of the cases are the same.
4. Fuzzy nano extremally disconnected space
In this section, we introduce fuzzy nano extremally disconnected space and we obtain several characterizations based on fuzzy set.
Definition 10. Let be a is called fuzzy nano (resp δ, δS, P and ) extremally disconnected (briefly, (resp. , , and )) space if the (resp. , , and ) closure of every (resp. , , and ) set in is (resp., , and ) set in , or equivalently, if the (resp. , and ) interior of every (resp. , and ) set of is (resp, , and ) set in .
Example 3. In Example 1, closure of every set in is set in .
Remark 3. Every space is (resp. , ) space.
Theorem 2. Let be a . Then the following are similar.(i) is space.(ii) is set, for each set of .(iii) for each set of .(iv) implies for each pair of set of .
Proof. (i) (ii) Let be a set of . Then is set of . Since is space, is set. But set . Therefore is set. (ii) (iii) Suppose that is a set of . Then . By assumption, is a set of .So, . (iii) (iv) Let and be sets of . We put . From the assumption, . (iv) (i) Let be a set of . Let . From the assumption, we obtain that . So, . Hence . Thus is a set of . Then is space.
Remark 4. The Theorem 2 also holds for and sets.
Theorem 3. Let be a is (resp. , , and ) space if and only if (resp. , and) for every (resp. , and ) set of .
Proof. Let be a set in a space. Then is set. This implies . Conversely, Let be a set and . Hence is set. Therefore is space.
The rest of the cases are the same.
5. Fuzzy nano normal spaces
In this section, we first present fuzzy nano normal spaces and scrutinize their essential properties.
Definition 11. Let be a is said to be fuzzy nano (resp. δ, δS, P and ) normal (briefly, (resp. and and and )) normal if for any two disjoint (resp. and and ) sets and , disjoint (resp and and and ) sets and .
Proposition 3. Every is .
Proof. Let be a . Then for any two disjoint sets and respectively, then there exists disjoint sets such that and . Since every sets are sets by Figure 1. Hence, is .
Theorem 4. In a , the following are comparable: (i) is . (ii) set in and every set containing , a set containing . (iii) For each pair of disjoint sets in , a set containing . (iv) For each pair of disjoint sets in , sets containing and respectively .
Proof. (i) (ii): Let be a set containing the set . Then is a set disjoint from . Since is , disjoint sets and containing and respectively. Then is disjoint from . Since if , the set is a set containing disjoint from . Hence . (ii) (iii): Let and be disjoint sets in . Then is a set containing . By (ii), there exists a set containing . Hence . This proves (iii).
(iii) (iv): Let and be disjoint sets in . Then, by (iii), there exists a set containing . Since is , and are disjoint sets in . Again by (iii), there exists a set containing . This proves (iv).
(iv) (i): Let and be the disjoint sets in . By (iv), sets and containing and respectively . Since and are disjoint sets containing and respectively. Thus is .
Theorem 5. Let be a is if and only if set set containing , there exists a set .
Proof. Let be . Let be a set and let be a set containing . Then are disjoint sets. Since is , disjoint sets and . Thus . Since is , so . Take .This implies that . Conversely, assume the situation remains the same. Let be two disjoint sets in . Then is a set containing . By assumption, there exists a set Since is and is . Then is . Now implies that Also . That is and are disjoint sets containing and respectively. This shows that is .
Theorem 6. For a , then the following are comparable: (i) is . (ii) For any two sets whose union is , subsets of of whose union is also .
Proof. (i) (ii): Let be two sets in a space . Then , are disjoint sets. Since is , then disjoint sets and . Let and . Then are subsets of respectively . This proves (ii). (ii) (i): Let be disjoint sets in . Then and are sets whose union is . By (ii), there exists sets and . Then are disjoint sets containing and respectively. Therefore is .
Theorem 7. Let be a function. (i) If is injective, , is then is . (ii) If is , and is then is .
Theorem 8. If given a pair of disjoint sets of , there is function and , then is .
Theorem 9. Let be a function. If is a , bijection of a space into a space and if every set in is , then is .
Proof. Let and be sets in . Then by assumption, is in . Since is a bijection, and is a set in . Since is , there exist disjoint sets and in and . Since is , and are disjoint sets in containing and respectively. Hence is .
Remark 5. Theorems 4, 5, 6, 7, 8 9 are also holds for and sets.
6. Strongly fuzzy nano normal spaces
The principles of strongly fuzzy nano normal spaces are introduced in this section. We describe each of these notions and show how they are related to one another.
Definition 12. A is said to be strongly fuzzy nano (resp δ, δS, P) normal (briefly, (resp. and and )) if for every pair of disjoint sets in , there are disjoint (resp. and ) sets and in containing respectively.
Example 4. In Example 1, , are sets. , are sets in containing respectively.
Theorem 10. Let be a . Every space is .
Proof. Suppose is . Let be disjoint sets in . Then are in . Since is , disjoint sets containing and respectively. Since, every is , and are in . This implies that is .
Theorem 11. In a , the following are comparable: (i) is . (ii) set in and every set containing , there exists a set containing . (iii) For each pair of disjoint sets in , there exists a set containing .
Proof. (i) (ii): Let be a set containing the set . Then is a set disjoint from . Since is , disjoint sets and containing respectively. Then is disjoint from , since if , the set is a set containing disjoint from . Hence . (ii) (iii): Let be disjoint sets in . Then is a set containing . By (ii), there exists a set containing . Hence . This proves (iii). (iii) (i): Let and be the disjoint sets in . By (iii), there exists a set containing . Take . Then and are disjoint sets containing and respectively. Thus is .
Theorem 12. For a , then the following are comparable: (i) is . (ii) For any two sets whose union is , subsets of and of whose union is also .
Proof. (i) (ii): Let be two sets in a space . Then are disjoint sets. Since is , then disjoint sets and and . Let and . Then are subsets of and respectively . This proves (ii). (ii) (i): Let be disjoint sets in . Then and are sets whose union is . By (ii), there exists sets and and . Then and are disjoint sets containing and respectively. Therefore is .
Theorem 13. Let be a function. (i) If is injective, , and is then is . (ii) If is , and is then is .
Proof. (i) Suppose is . Let and be disjoint sets in . Since is , and are in . Since is , disjoint sets and in and . Now and . Since is a map, and are in . Also and is injective, then . Thus and are disjoint sets in containing and respectively. Thus, is . (ii) Suppose is . Let and be disjoint sets in . Since is and , and are in . Since is , there exist disjoint sets and in and . That is and . Since is , and are disjoint and . Thus is .
Remark 6. Theorems 10, 11, 12 13 are also holds for and sets.
7. Fuzzy score function
We provide a fuzzy scoring function for decision-making problems using fuzzy information in this part, which is based on a methodical approach.
Definition 13. Let . The Fuzzy score function (in short, ) is that represents the average of positiveness of the fuzzy component .
The specific technique to deal with selecting the correct qualities and alternatives in a decision-making situation utilising fuzzy sets is proposed in the following fundamental steps. Step 1: Problem field selection: Consider the universe of discourse (set of objects) , the set of alternatives , the set of decision attributes . Step 2: Construct a fuzzy matrix of alternative verses objects and object verses decision attributes. Calculation Part: Step 3: Frame the in-discernibility relation on . Step 4: Construct the fuzzy nano topologies and . Step 5: Find the score values by Definition 1 each of the entries of the . Conclusion part: Step 6: Organize the fuzzy score values of the alternatives and the attributes . Choose the attribute for the alternative and for the alternative etc. If , then ignore , where .
7.1. Numerical example
New medical breakthroughs have expanded the number of data available to clinicians, which includes vulnerabilities. The process of grouping multiple sets of symptoms under a single term of illness is extremely challenging in medical diagnosis. In this section, we use a medical diagnosis problem to demonstrate the usefulness and applicability of the above-mentioned approach. Step 1: Problem field selection: Consider the following tables, which provide information from five patients who were consulted by physicians, Patient 1 (Pat 5), Patient 2 (Pat 5), Patient 3 (Pat 5), Patient 4 (Pat 5), Patient 5 (Pat 5) and symptoms are Weight gain (Wg), Tiredness (Td), Myalgia (Ml), Swelling of legs (Sl), Mensus Problem (Mp). We need to find the patient and to find the disease such as Lymphedema, Insomnia, Hypothyroidism, Menarche, Arthritis of the patient. The data in Tables 1 and 2 are explained by the membership, the indeterminacy and the non-membership functions of the patients and diseases respectively. Step 2: Construct the in-discernibility relation for the correlation between the symptoms is given as ,. Step 3: From fuzzy nano topologies for and :(i).(ii).(iii).(iv).(v).(i).(ii).(iii).(iv).(v). Step 5: Find fuzzy score functions: (i) . (ii) . (iii) . (iv) . (v) . (i) . (ii) . (iii) . (iv) . (v) . Step 6: Final decision: Arrange fuzzy nano score values for the alternatives , , , , and the attributes , , , , in ascending order. We get the following sequences and . Thus the patient Pat 5 suffers from Hypothyroidism, the patient Pat5 suffers from Arthritis, the patient Pat5 suffers from Menarche, the patient Pat5 suffers from Lymphedema and the patient Pat5 suffers from Insomnia. The results are presented in Figures 2 and 3.
8. Final thoughts and future work
This paper adds to the growing body of knowledge about fuzzy nano topological spaces. The obtained results show that most of the offered concepts’ nano topological features are kept in the framework of fuzzy nano topologies, implying that some topological prerequisites are unnecessary. Because the study’s limitations are relaxed, exploring nano topological notions using fuzzy nano topologies has a benefit. On the other hand, by extending fuzzy nano locally closed sets, a few characteristics of particular topological concepts are partially lost. We will finish introducing the main fuzzy nano topological concepts using fuzzy nano open sets, such as fuzzy nano locally continuous, respective mappings and homomorphisms, separation axioms, compactness and connectedness in fuzzy nano topological spaces, in this work. Our study plan also includes testing the concepts and results presented here with various generalisations of fuzzy nano open sets, such as fuzzy nano open and fuzzy nano open sets. Furthermore, we will use these expansions of fuzzy nano open sets to present new types of rough approximations and apply them to improve set accuracy metrics.
Data Availability
Data used to support this study are included within this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.