Fuzzy Nano z -locally Closed Sets, Extremally Disconnected Spaces, Normal Spaces, and Their Application

In this paper, we introduce fuzzy nano (resp. δ , δ S, P and Z) locally closed set and fuzzy nano (resp. δ , δ S, P and Z) extremally disconnected spaces in fuzzy nano topological spaces. Also, we introduce some new spaces called fuzzy nano (resp. δ , δ S, P and Z) normal spaces and strongly fuzzy nano (resp. δ , δ S, P and Z) normal spaces with the help of fuzzy nano (resp. δ , δ S, 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.


Introduction
rough his signi cant theory on fuzzy sets, Zadeh [1] made the rst e ective attempt in mathematical modeling to contain non-probabilistic uncertainty, i.e. uncertainty that is not caused by randomness of an event. e study of fuzzy calculus plays a vital role in the eld of mathematics due to its useful applications in variety of scienti c 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 rst 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. is 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. e 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 ivagar [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. e Nano open sets are the constituents of a Nano topological space, while the Nano closed sets are their complements. e 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 delta open sets in nano topological space were investigated.
is paper follows the definition of Lellis ivagar et al. [9]. Generalizations of (fuzzy nano) open sets are a major topic in (fuzzy nano) topology. One of the important generalizations is a Z-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 M-open sets [11], nano Z-open sets [12], Z-closed sets in double fuzzy topological spaces [13,14] and Z-open sets in a fuzzy nano topological spaces by angammal et al. [15]. Kuratowski and Sierpinski [16] explored the difference of two closed subsets of a n-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 LC-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 reallife 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. e 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. ese 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. e problems associated with these cases are interesting, and developing a hypothesis for them has prompted many scholars [19][20][21] to pay attention to them Motivation and objective. No investigation on fuzzy nano Z locally closed set, fuzzy nano Z extremally disconnected spaces, fuzzy nano Z normal spaces and strongly fuzzy nano Z 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]. e 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 FNts. Our proposed model and techniques are discussed in terms of their robustness, durability, superiority, and simplicity. is 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. e 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 Z locally closed set and establish some of characterizes. e concept of fuzzy nano Z 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 Z normal space and strongly fuzzy nano Z 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 FNts. 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.

Preliminaries
is part explains the concepts and findings that we need to know in order to comprehend the manuscript.
Definition 1 (see [1]). A function f from X into the unit interval I is called a fuzzy set (briefly, Fs) in X.
Definition 2 (see [1]). If G and H are any two fuzzy subsets (briefly, Fsubs) of a set X, then

Advances in Fuzzy Systems
Definition 3 (see [1]). e complement of a Fsubs G in X, denoted by 1 − G, is the Fsubs of X defined by 1 − G(l), ∀ l in X.
Definition 4 (see [9]). Let U be a non-empty set and R be an equivalence relation on U. Let F be a Fs in U with the membership function μ F . e fuzzy nano lower (upper) approximations and fuzzy nano boundary of F in the approximation (U, R) denoted by FN (F), FN(F) and B FN (F) are respectively defined as follows:

Fuzzy nano Z locally closed sets
e idea of fuzzy nano Z locally closed sets, which represents a class of generalisations of fuzzy nano Z open sets, is introduced in this section. e main features of fuzzy nano Z closed sets are established, as well as certain characterizations.
Definition 5. Let (U, τ F (F)) be a FNts with respect to F where F is a fuzzy subset of U. Let S be a fuzzy subset of U.
en fuzzy nano (ii) closure of S (briefly, FNcl(S)) is represented as e complement of an FNδScl(S) (resp. FℵZO(U, A)) set is called a fuzzy nano δ (resp. fuzzy nano δ-semi & fuzzy nano pre) closed (briefly, FNδc (resp. FℵZO(U, A)) in U. Definition 6. Let (U, τ F (F)) be a FNts. en a fuzzy subset S in U is said to be a fuzzy nano intersection) of all FℵZo (resp. FℵZc) sets contained in O and denoted by FℵZint(O) (resp. FℵZcl(O)).
(1)    e converse of the preceding proposition does not have to be true, as the following example demonstrates.
□ e rest of the cases are the same.

Fuzzy nano Z extremally disconnected space
In this section, we introduce fuzzy nano Z extremally disconnected space and we obtain several characterizations based on fuzzy set.

Fuzzy nano Z normal spaces
In this section, we first present fuzzy nano Z normal spaces and scrutinize their essential properties. Definition 11. Let (U, τ F (F)) be a FNts is said to be fuzzy nano (resp. δ, δS, P and Z) normal (briefly, FNNor (resp.FNZNor and FNZC and FNZc and FNZNor)) normal if for any two disjoint FNc (resp. FNZNor and   Proof. Let (U, τ F (F)) be FNZNor. Let F be a FNZc set and let G be a FNZo set containing F.

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at is M and (FNcl(M)) c are disjoint FNZo sets containing H 1 and H 2 respectively. is shows that (U, τ F (F)) is FNZNor. □ Theorem 6. For a FNts (U, τ F (F)), then the following are comparable: FNZc sets in U. en A c and B c are FNZo sets whose union is 1 N . By (ii), there exists FNZc sets Theorem 9. Let f: (U 1 , τ F (F 1 )) ⟶ (U 2 , τ F (F 2 )) be a function. If f is a FNCts, FNZO bijection of a FNNor space U 1 into a space U 2 and if every FNZc set in U 2 is FNc, then U 2 is FNZReg.
Proof. Let M 1 and M 2 be FNZc sets in U 2 . en by assumption, M 2 is FNc in U 2 . Since f is a FNCts bijection, f − 1 (M 1 ) and f − 1 (M 2 ) is a FNc set in U 1 . Since U 1 is FNNor, there exist disjoint FNo sets L 1 and L 2 in

Strongly fuzzy nano Z normal spaces
e principles of strongly fuzzy nano Z normal spaces are introduced in this section. We describe each of these notions and show how they are related to one another.

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. e 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) m, the set of alternatives n, the set of decision attributes p.
Step 2: Construct a fuzzy matrix of alternative verses objects and object verses decision attributes. Calculation Part: Step 3: Frame the in-discernibility relation R on m.
Step 4: Construct the fuzzy nano topologies τ j and ] k .
7.1. Numerical example. New medical breakthroughs have expanded the number of data available to clinicians, which includes vulnerabilities. e 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. e data in Tables 1 and 2 are explained by the membership, the indeterminacy and the non-membership functions of the patients and diseases respectively.

Final thoughts and future work
is paper adds to the growing body of knowledge about fuzzy nano topological spaces. e 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 Z 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 Z open sets, such as fuzzy nano Z 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 Z open sets, such as fuzzy nano e open and fuzzy nano Z * open sets. Furthermore, we will use these expansions of fuzzy nano Z 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.