# Fuzzy Nanoz-locally Closed Sets, Extremally Disconnected Spaces, Normal Spaces, and Their Application

Authors: R. Thangammal; M. Saraswathi; A. Vadivel; Samad Noeiaghdam; C. John Sundar; V. Govindan; Aiyared Iampan
Journal: Advances in Fuzzy Systems
 (2022)

---

## 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.


---

## Body

## 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 delta 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 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 Thangammal 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 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 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]. 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 FNts. 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: Section2 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. The 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.


## 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 functionf from X into the unit interval I is called a fuzzy set (briefly, ℱs) in X.Definition 2 (see [1]).
IfG and H are any two fuzzy subsets (briefly, ℱsubs) of a set X, then
(i)G≤H iff μGl≤μHl, ∀l in X. (ii) G=H, if Gl=Hl∀l in X. (iii) G∨Hl=maxGl,Hl, ∀l in X. (iv) G∧Hl=minGl,Hl, ∀l in X.Definition 3 (see [1]).
The complement of aℱsubsG in X, denoted by 1−G, is the ℱsubs of X defined by 1−Gl, ∀l in X.Definition 4 (see [9]).
LetU be a non-empty set and R be an equivalence relation on U. Let F be a ℱs in U with the membership function μF. The fuzzy nano lower (upper) approximations and fuzzy nano boundary of F in the approximation U,R denoted by FN¯F,FN¯F and BFNF are respectively defined as follows: (i) FN¯F=l,μR¯AL/Y∈lR,l∈U (ii) FN¯F=l,μR¯Al/y∈lR,l∈U (iii) BFNF=FN¯F−FN¯F where μR¯Al=∧y∈lRμAy. μR¯Al=∨y∈lRμAy. The collection τℱF=0ℱ,1ℱ,FN¯F,FN¯F,BFNF forms a topology called as fuzzy nano topology and U,τℱF as fuzzy nano topological space (briefly, FNts). The elements of τℱF are called fuzzy nano open (briefly, FNo) sets. Elements of τℱFc are called fuzzy nano closed (briefly, FNc) sets.


## 3. Fuzzy nanoZ locally closed sets

The idea of fuzzy nanoZ locally closed sets, which represents a class of generalisations of fuzzy nano Z open sets, is introduced in this section. The main features of fuzzy nano Z closed sets are established, as well as certain characterizations.Definition 5.
LetU,τℱF be a FNts with respect to F where F is a fuzzy subset of U. Let S be a fuzzy subset of U. Then fuzzy nano(i)
interior ofS (briefly, FNintS) is represented as FNintS=∨O:O≤S&Ois aFNoset inU.(ii)
closure ofS (briefly, FNclS) is represented as FNclS=∧L:S≤L&Lis aFNcset inU.(iii)
regular open (briefly,FNro) set if S=FNintFNclS.(iv)
regular closed (briefly,FNrc) set if S=FNclFNintS.(v)
δ interior of S (briefly, FNδintS) is represented as FNδintS=∨O:O≤S&Ois aFNroset inU.(vi)
δ closure of S (briefly, FNδclS) is represented as FNδclS=∧L:S≤L&Lis aFNrcset inU.(vii)
δ-open (briefly, FNδo) set if S=FNδintS.(viii)
δ-semi open (briefly, FNδSo) set if S≤FNclFNδintS.(ix)
pre open (briefly,FNPo) set if S≤FNintFNclS.(x)
δ semi interior of S (briefly, FNδSintS) is represented as FNδSintS=∨O:O≤S&Ois aFNδSoset inU.(xi)
δ semi closure of S (briefly, FNδSclS) is represented as FNδSclS=∧L:S≤L&Lis aFNδSset inU.(xii)
pre interior ofS (briefly, FNPint(S)) is represented as FNδSintS=∨O:O≤S&Ois aFNδSoset inU.(xiii)
pre closure ofS (briefly, FNPint(S)) is represented as FNδSintS=∨O:O≤S&Ois aFNδSoset inU.
The complement of anFNδSclS (resp. ℱℵZOU,A) set is called a fuzzy nano δ (resp. fuzzy nano δ-semi & fuzzy nano pre) closed (briefly, FNδc (resp. ℱℵZOU,A) in U. Definition 6. Let U,τℱF be a FNts. Then a fuzzy subset S in U is said to be a fuzzy nano(i)
Z-open (briefly, ℱℵZo) set if S≤ℱℵclℱℵδintS∨ℱℵintℱℵclS,(ii)
Z-closed (briefly, ℱℵZc) set if ℱℵintℱℵδclS∧ℱℵclℱℵintS≤S.(iii)
Z-interior (resp. closure) of O is the union (resp. intersection) of all ℱℵZo (resp. ℱℵZc) sets contained in O and denoted by ℱℵZintO (resp. ℱℵZclO).
AllℱℵZo (resp. ℱℵZc) sets of a space U,τℱF will be denoted by ℱℵZOU,A (resp. ℱℵZCU,A).Remark 1.
The following diagram shows the relationship between any set inℱℵanotsts of FNZos’s (resp. FNZcs’s).Definition 7.
A functionh:U1,τℱF1⟶U2,τℱF2 is said to be fuzzy(i)
nano (resp.Z) continuous (briefly, ℱℵCts (resp. ℱℵZCts)), if ∀ℱℵo set M of U2, the set h−1M is ℱℵo (resp. ℱℵZo) set of U1.(ii)
nanoZ irresolute (briefly, ℱℵZIrr) function, if ∀ℱℵZo subset M of U2, the set h−1M is ℱℵZo subset of U1.(iii)
nano (resp.δ, δS, P and Z) open map (briefly, FNO and FNZO)) if the image of each FNo set in U1 is FNo and FNZo) in U2.(iv)
nano (resδ, δS, P and Z) closed map (briefly, FNC and FNZC)) if the image of each FNc set in U1 is FNc and FNZc) in U2.Definition 8.
LetA and B be any two fuzzy subsets of a FNts’s. Then A is fuzzy nan δ, δS, P and Z) q-neighbourhood (briefly, FNq-nbhd (resp. FNδq, FNδS, FNδSo, FNZo anbhd and FNZq-nbhd)) with B if there exists a FNo and FNZo) set O with AqO≤B.Definition 9.
LetU,τℱF be a FNts is called fuzzy nano (resp. δ, δS, P and Z) locally closed (briefly, FNLC (respFNδS, FNδSo, FNZo a and FNZLC)) set if A=B∧C where B is a FNo (resp. FNδS, FNδSo, FNZo and FNZo) set and C is a FNc (resp. FNδS, FNδSo, FNZo a and FNZc) set.Example 1.
AssumeU=s1,s2,s3,s4 and U/R=s1,s4,s2,s3.
LetS=s1/0.2,s2/0.3,s3/0.4,s4/0.1 be a ℱsubs of U.(1)FN¯S=s1,s40.1,s20.3,s30.4,FN¯S=s1,s40.2,s20.3,s30.4,BFNS=s1,s40.2,s20.3,s30.4.
ThusτℱS=0ℱ,1ℱ,FN¯S,FN¯S=BFNS.
Then(i)
FNLC⇒C=A∧B=s1,s4/0.1,s2/0.3,s3/0.4∧s1,s4/0.8,s2/0.7,s3/0.6==s1,s4/0.1,s2/0.3,s3/0.4(ii)
FNδSLC⇒C=A∧B=s1,s4/0.4,s2/0.6,s3/0.5∧s1,s4/0.3,s2/0.7,s3/0.4=s1,s4/0.3,s2/0.6,s3/0.4(iii)
FNZLC⇒C=A∧B=s1,s4/0.4,s2/0.6,s3/0.5∧s1,s4/0.3,s2/0.7,s3/0.4=s1,s4/0.3,s2/0.6,s3/0.4.(iv)
FNδSLC⇒C=A∧B=s1,s4/0.4,s2/0.6,s3/0.5∧s1,s4/0.3,s2/0.7,s3/0.4=s1,s4/0.3,s2/0.6,s3/0.4.(v)
FNZLC⇒C=A∧B=s1,s4/0.4,s2/0.6,s3/0.5∧s1,s4/0.3,s2/0.7,s3/0.4=s1,s4/0.3,s2/0.6,s3/0.4.Proposition 1.
LetU,τℱF be a FNts. (i) Every FNZc (resp. FNZo) set is FNZLC set. (ii) Every FNc (resp. FNo) set is FNLC set. (iii) Every FNδc (resp. FNδo) set is FNδLC set. (iv) Every FNδSc (resp. FNδSo) set is FNδSLC set. (v) Every FNδSc (resp. FNδSo) set is FNδSLC set.Proof.
(i) LetA be a FNZc set in U. Then A can be written as A=A∧1N, where A is a FNZLC set and 1N is a FNZo set. Therefore A is a FNZc set. The rest of the cases are the same.Proposition 2.
LetU,τℱF be a FNts. Every FNLC set is FNZLC (resp. FNLC and FNZLC) set.Proof.
LetA be a FNLC set in U. Then A can be written as A=U∧V, where U is FNo set and V is FNc set. Since every FNoFNc set is FNZoFNZc, A is the intersection of FNZo set and FNZc set and hence A is FNZLC 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 Example1, C=A∧B=s1,s4/0.4,s2/0.6,s3/0.5∧s1,s4/0.3,s2/0.7,s3/0.4=s1,s4/0.3,s2/0.6,s3/0.4. Then C is FNZLC but not FNLC.Theorem 1.
LetU,τℱF be a FNts. Then O is FNZLC (resp. FNLC,Every FNδSc (resp. FNδSo) set is FNδSLC) if and only if O=B∧FNZclO (resp. O=B∧FNclO, O≤B∧FNZclO and therefore O=B∧FNZclO for some FNZo) for some FNZo (resp. FNo,O≤FNZclO implies O≤B∧FNZclO) set B.Proof.
LetO be a FNZLC set. Then O=B∧C, where B is FNZo set and C is FNZc set in X. Since O≤C,FNZclO≤FNZclC and so B∧FNZclO≤O. Olso O≤B and O≤FNZclO implies O≤B∧FNZclO and therefore O=B∧FNZclO for some FNZo set B. Conversely, assume O=B∧FNZclO. Since FNZclO is FNZc set and B is FNZo set, O is FNZc set.The rest of the cases are the same.


## 4. Fuzzy nanoZ extremally disconnected space

In this section, we introduce fuzzy nanoZ extremally disconnected space and we obtain several characterizations based on fuzzy set.Definition 10.
LetU,τℱF be a FNts is called fuzzy nano (resp δ, δS, P and Z) extremally disconnected (briefly, FNExDCon (resp. FNExDCon, FNExDCon, FNZExDCon and FNZExDCon)) space if the FN (resp. FNts, FNZExDCon, FNZo and FNZ) closure of every FNo (resp. FNts, FNZExDCon, FNZo and FNZo) set in U is FNo (resp.FNts, FNZExDCon, FNZo and FNZo) set in U, or equivalently, if the FN (resp. FNZExDCon, FNZo and FNZ) interior of every FNc (resp. FNZExDCon, FNZo and FNZc) set of U is FNc (respFNts, FNZExDCon, FNZo and FNZc) set in U.Example 3.
In Example1, FNZ closure of every FNZo set in U is FNZo set in U.Remark 3.
EveryFNExDCon space is FNZExDCon (resp. FNZExDCon, FNZo) space.Theorem 2.
LetU,τℱF be a FNts. Then the following are similar.(i)
U is FNZExDCon space.(ii)
FNZintA is FNZc set, for each FNZc set of U.(iii)
FNZclFNZclAc=FNZclAc for each FNZo set of U.(iv)
B=FNZclAc implies FNZclB=FNZclAc for each pair of FNZo set A&B of U.Proof.
(i)⇒ (ii) Let A be a FNZc set of U. Then Ac is FNZo set of U. Since U is FNZExDCon space, FNZclAc is FNZo set. But FNZo set FNZclAc=FNZintAc. Therefore FNZintA is FNZc set. (ii) ⇒ (iii) Suppose that A is a FNZo set of U. Then FNZclFNZclAc=FNZclFNZintAc. By assumption, FNZintAc is a FNZc set of U.So, FNZclFNZintAc=FNZintAc=FNZclAc. (iii) ⇒ (iv) Let A and B be FNZo sets of U. We put B=FNZclAc. From the assumption, FNZclB=FNZclFNZclAc=FNZclAc. (iv) ⇒ (i) Let A be a FNZo set of U. Let B=FNZclAc. From the assumption, we obtain that FNZclB=FNZclAc. So, FNZclBc=FNZclA. Hence FNZintBc=FNZclA. Thus FNZclA is a FNZo set A of U. Then U is FNZExDCon space.Remark 4.
The Theorem2 also holds for FNZNor and FNZC sets.Theorem 3.
LetU,τℱF be a FNts is FNZExDCon (resp. FNExDCon, FNExDCon, FNExDCon and FNExDCon) space if and only if FNZclA=FNZintFNZclA (resp. FNZclB=FNZclFNZclAc=FNZclAc, FNZclFNZintAc=FNZintAc=FNZclAc andFNZclB=FNZclFNZclAc=FNZclAc) for every FNZo (resp. FNo, FNZNor and FNZC) set A of U.Proof.
LetA be a FNZo set in a FNZExDCon space. Then FNZclA is FNZo set. This implies FNZclA=FNZintFNZclA. Conversely, Let A be a FNZo set and FNZclA=FNZintFNZclA. Hence FNZclA is FNZo set. Therefore U is FNZExDCon space.The rest of the cases are the same.


## 5. Fuzzy nanoZ normal spaces

In this section, we first present fuzzy nanoZ normal spaces and scrutinize their essential properties.Definition 11.
LetU,τℱF 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 FNZC and FNZc) sets A and B, ∃ disjoint FNo (respFNZNor and FNZC and FNZc and FNZo) sets L&M∋A≤L and B≤M.Proposition 3.
EveryFNNor is FNZNor.Proof.
LetU be a FNNor. Then for any two disjoint FNc sets A and B respectively, then there exists disjoint FNo sets L&M such that A≤L and B≤M. Since every FNo sets are FNZo sets by Figure 1. Hence, U is FNZNor.Figure 1
FNZos’s in FNts.Theorem 4.
In aFNtsU,τℱF, the following are comparable: (i) U is FNZNor. (ii) ∀FNZc set A in U and every FNZo set L containing A, ∃ a FNZo set M containing A∍FNclM≤L. (iii) For each pair of disjoint FNZc sets A&B in U, ∃ a FNZo set L containing A∑FNclL∧B=0N. (iv) For each pair of disjoint FNZc sets A∋B in U, ∃FNZo sets L&M containing A and B respectively ∑FNclL∧FNclM=0N.Proof.
(i)⇒ (ii): Let L be a FNZo set containing the FNZc set A. Then B=Lc is a FNZc set disjoint from A. Since U is FNZNor, ∃ disjoint FNZo sets M and W containing A and B respectively. Then FNclM is disjoint from B. Since if yβ∈B, the set W is a FNZo set containing yβ disjoint from M. Hence FNclM≤L. (ii) ⇒ (iii): Let A and B be disjoint FNZc sets in U. Then Bc is a FNZo set containing A. By (ii), there exists a FNZo set L containing A∑FNclL≤Bc. Hence FNclL∧B=0N. This proves (iii).(iii)⇒ (iv): Let A and B be disjoint FNZc sets in U. Then, by (iii), there exists a FNZo set L containing A∑FNclL≤Bc. Since FNclL is FNZc, B and FNclL are disjoint FNZc sets in U. Again by (iii), there exists a FNZo set M containing ∑FNclL∧FNclM=0N. This proves (iv).(iv)⇒ (i): Let A and B be the disjoint FNZc sets in U. By (iv), ∃FNZo sets L and M containing A and B respectively ∑FNclL∧FNclM=0N. Since L∧M≤FNclL∧FNclM,L and M are disjoint FNZo sets containing A and B respectively. Thus U is FNZNor.Theorem 5.
LetU,τℱF be a FNts is FNZNor if and only if ∀FNZc set F&FNZo set G containing F, there exists a FNZo set M∑F≤M≤FNclM≤G.Proof.
LetU,τℱF be FNZNor. Let F be a FNZc set and let G be a FNZo set containing F. Then F&Gc are disjoint FNZc sets. Since U is FNZNor, ∃ disjoint FNZo sets M1&M2∋F≤M1 and Gc≤M2. Thus F≤M1≤M2c≤G. Since M2c is FNZc, so FNclM≤FNclM2c=M2c≤G. Take M=M1.This implies that F≤M≤FNclM≤G. Conversely, assume the situation remains the same. Let H1&H2 be two disjoint FNZc sets in U. Then H2c is a FNZo set containing H1. By assumption, there exists a FNZo set M∑H1≤M≤FNclM≤H2c Since M is FNZo and FNclM is FNZc. Then FNclMc is FNZo. Now FNclM≤H2c implies that H2≤FNclMc Also M∧FNclMc≤FNclM∧FNclMc=0N. That is M and FNclMc are disjoint FNZo sets containing H1 and H2 respectively. This shows that U,τℱF is FNZNor.Theorem 6.
For aFNtsU,τℱF, then the following are comparable: (i) U is FNZNor. (ii) For any two FNZo sets L&M whose union is 1N, ∃FNZc subsets A of L&B of M whose union is also U.Proof.
(i)⇒ (ii): Let L&M be two FNZo sets in a FNZNor space U∋1N=L∨M. Then Lc, Mc are disjoint FNZc sets. Since U is FNZNor, then ∃ disjoint FNZo sets G1&G2∋Lc≤G1 and Mc≤G2. Let A=G1c and B=G2c. Then A&B are FNZc subsets of L&M respectively ∋A∨B=1N. This proves (ii). (ii) ⇒ (i): Let A&B be disjoint FNZc sets in U. Then Ac and Bc are FNZo sets whose union is 1N. By (ii), there exists FNZc sets F1&F2∋F1≤Ac,F2≤Bc and F1∨F2=1N. Then F1c&F2c are disjoint FNZo sets containing A and B respectively. Therefore U is FNZNor.Theorem 7.
Letf:U1,τℱF1⟶U2,τℱF2 be a function. (i) If f is injective, FNZIrr, FNZo&U1 is FNZNor then U2 is FNZNor. (ii) If f is FNZIrr, FNZc and U2 is FNZNor then U1 is FNZNor.Theorem 8.
If given a pair of disjointFNZc sets A,B of U,τℱF, there is FNZCts function f∑fA=0N and fB=1N, then U,τℱF is FNZNor.Theorem 9.
Letf:U1,τℱF1⟶U2,τℱF2 be a function. If f is a FNCts, FNZO bijection of a FNNor space U1 into a space U2 and if every FNZc set in U2 is FNc, then U2 is FNZReg.Proof.
LetM1 and M2 be FNZc sets in U2. Then by assumption, M2 is FNc in U2. Since f is a FNCts bijection, f−1M1 and f−1M2 is a FNc set in U1. Since U1 is FNNor, there exist disjoint FNo sets L1 and L2 in U1∑f−1M1≤L1 and f−1M2≤L2. Since f is FNZO, fL1 and fL2 are disjoint FNZo sets in U2 containing M1 and M2 respectively. Hence U2 is FNZNor.Remark 5.
Theorems4, 5, 6, 7, 8 & 9 are also holds for FNZNor and FNZC sets.


## 6. Strongly fuzzy nanoZ normal spaces

The principles of strongly fuzzy nanoZ normal spaces are introduced in this section. We describe each of these notions and show how they are related to one another.Definition 12.
AFNtsU,τℱF is said to be strongly fuzzy nano Z (resp δ, δS, P) normal (briefly, StFNZNor (resp. FNZNor and FNZC and FNZc)) if for every pair of disjoint FNc sets A&B in U, there are disjoint FNZo (resp. FNZNor and FNZC) sets L and M in U containing A&B respectively.Example 4.
In Example1, A=s1,s4/0.8,s2/0.7,s3/0.6, B=s1,s4/0.9,s2/0.7,s3/0.6 are FNc sets. L=s1,s4/0.8,s2/0.7,s3/0.7, M=s1,s4/0.9,s2/0.7,s3/0.7 are FNZc sets in U containing A&B respectively.Theorem 10.
LetU,τℱF−b±b2−4ac/2a be a FNts. Every FNZNor space is StFNZNor.Proof.
SupposeU is FNZNor. Let A&B be disjoint FNc sets in U. Then A&B are FNZc in U. Since U is FNZNor, ∃ disjoint FNo sets L&M containing A and B respectively. Since, every FNo is FNZo, L and M are FNZo in U. This implies that U is StFNZNor.Theorem 11.
In aFNtsU,τℱF, the following are comparable: (i) U is StFNZNor. (ii) ∀FNc set F in U and every FNo set L containing F, there exists a FNZo set M containing F∑FNZclM≤L. (iii) For each pair of disjoint FNc sets M1&M2 in U, there exists a FNZo set L containing M1∑FNZclL∧M2=0N.Proof.
(i)⇒ (ii): Let L be a FNo set containing the FNc set F. Then H=Lc is a FNc set disjoint from F. Since U is StFNZNor, ∃ disjoint FNZo sets M and W containing F&H respectively. Then FNZclM is disjoint from H, since if yβ∈H, the set W is a FNZo set containing yβ disjoint from M. Hence FNZclM≤L. (ii) ⇒ (iii): Let M1&M2 be disjoint FNc sets in U. Then M2c is a FNo set containing M1. By (ii), there exists a FNZo set L containing M1∑FNZclL≤M2c. Hence FNZclL∧M2=0N. This proves (iii). (iii) ⇒ (i): Let M1 and M2 be the disjoint FNZc sets in U. By (iii), there exists a FNZo set L containing M1∑FNZclL∧M2=0N. Take M=FNZclLc. Then L and M are disjoint FNZo sets containing M1 and M2 respectively. Thus U is StFNZNor.Theorem 12.
For aFNtsU,τℱF, then the following are comparable: (i) U is StFNZNor. (ii) For any two FNo sets L&M whose union is 1N, ∃FNZc subsets M1 of L and M2 of M whose union is also 1N.Proof.
(i)⇒ (ii): Let L&M be two FNo sets in a StFNZNor space U∋1N=L∨M. Then Lc,Mc are disjoint FNc sets. Since U is StFNZNor, then ∃ disjoint FNZo sets G1 and G2∋Lc≤G1 and Mc≤G2. Let M1=G1c and M2=G2c. Then M1&M2 are FNZc subsets of L and M respectively ∋M1∨M2=1N. This proves (ii). (ii) ⇒ (i): Let M1&M2 be disjoint FNc sets in U. Then M1c and M2c are FNo sets whose union is U. By (ii), there exists FNZc sets F1 and F2∋F1≤M1c,F2≤M2c and F1∨F2=1N. Then F1c and F2c are disjoint FNZo sets containing M1 and M2 respectively. Therefore U is StFNZNor.Theorem 13.
Leth:U1,τℱF1⟶U2,τℱF2 be a function. (i) If h is injective, FNCts, FNZO and U1 is StFNZNor then U2 is StFNZNor. (ii) If h is FNZIrr, FNZO and U2 is StFNZNor then U1 is StFNZNor.Proof.
(i) SupposeU1 is StFNZNor. Let M1 and M2 be disjoint FNc sets in U2. Since h is FNCts, h−1M1 and h−1M2 are FNc in U1. Since U1 is StFNZNor, ∃ disjoint FNZo sets L and M in U1∑h−1M1≤L and h−1M2≤M. Now h−1M1≤L⇒M1≤hL and h−1M2≤M⇒M2≤hM. Since h is a FNZO map, hL and hM are FNZo in U2. Also L∧M=0N⇒hL∧M=0N and h is injective, then hL∧hM=0N. Thus hL and hM are disjoint FNZo sets in U2 containing M1 and M2 respectively. Thus, U2 is StFNZNor. (ii) Suppose U2 is FNZNor. Let M1 and M2 be disjoint FNc sets in U1. Since h is FNZIrr and FNZC, hM1 and hM2 are FNZc in U2. Since U2 is FNZNor, there exist disjoint FNZo sets L and M in U2∑hM1≤L and hM2≤M. That is M1≤h−1L and M2≤h−1M. Since h is FNZIrr, h−1L and h−1M are disjoint FNZo∑M1≤h−1L and M2≤h−1M. Thus U1 is FNZNor.Remark 6.
Theorems10, 11, 12 & 13 are also holds for FNZNor and FNZC 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.
LetS:M⟶0,1. The Fuzzy score function (in short, ℱSF) is SM=1/k∑i=1kμMi that represents the average of positiveness of the fuzzy component μM.
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)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 relationR on m.Step 4: Construct the fuzzy nano topologiesτj and νk.Step 5: Find the score values by Definition1 each of the entries of the FNts. Conclusion part:Step 6: Organize the fuzzy score values of the alternativesτ1≤τ2≤⋯≤τn and the attributes ν1≤ν2≤⋯≤νp. Choose the attribute νp for the alternative τ1 and νp−1 for the alternative τ2 etc. If n<p, then ignore νk, where k=1,2,…,n−p.


### 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 (Pat5), 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 asU/R=Wg,Ml,Td,Sl,Mp,.Step 3: From fuzzy nano topologies forτj and νk:(i)
τ1∗=0ℱ,1ℱ,0.9,0.3,0.2.(ii)
τ2∗=0ℱ,1ℱ,0.8,0.1,0.4,0.3.(iii)
τ3∗=0ℱ,1ℱ,0.8,0.3,0.2,0.4.(iv)
τ4∗=0ℱ,1ℱ,0.3,0.1,0.2,0.4,0.9.(v)
τ5∗=0ℱ,1ℱ,0.3,0.6,0.4,0.7.(i)
ν1∗=0ℱ,1ℱ,0.2,0.7,0.9.(ii)
ν2∗=0ℱ,1ℱ,0.9,0.2.(iii)
ν3∗=0ℱ,1ℱ,0.9,0.1,0.2.(iv)
ν4∗=0ℱ,1ℱ,0.6,0.1,0.2,0.9.(v)
ν5∗=0ℱ,1ℱ,0.1,0.9,0.4,0.3.Step 5: Find fuzzy score functions: (i)ℱSFτ1=0.48. (ii) ℱSFτ2=0.4333. (iii) ℱSFτ3=0.45. (iv) ℱSFτ4=0.4143. (v) ℱSFτ5=0.5.(i)ℱSFν1=0.56. (ii) ℱSFν2=0.525. (iii) ℱSFν3=0.44. (iv) ℱSFν4=0.4667. (v) ℱSFν5=0.45.Step 6: Final decision: Arrange fuzzy nano score values for the alternativesτ1, τ2, τ3, τ4, τ5 and the attributes ν1, ν2, ν3, ν4, ν5 in ascending order. We get the following sequences τ5≤τ1≤τ3≤τ2≤τ4 and ν1≤ν2≤ν4≤ν5≤ν3. 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.Table 1
Fuzzy values for patients.
Symptoms/PatientPat1Pat2Pat3Pat4Pat5Weight gain0.90.80.00.30.3Tiredness0.00.10.80.10.6Myalgia0.30.80.30.20.3Swelling of legs0.90.40.20.40.4Mensus Problem0.20.30.40.90.7Table 2
Fuzzy values for disease.
Symptoms/patientsWeight gainTirednessMyalgiaSwelling of legsMensus problemLymphedema0.00.20.70.90.2Insomnia0.00.90.20.20.2Hypothyroidism0.90.10.00.10.2Menarche0.60.10.20.20.9Arthritis0.00.10.90.40.3Figure 2
Fuzzy score values for diseases.Figure 3
Fuzzy Score values for Patients.


## 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 (Pat5), 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 asU/R=Wg,Ml,Td,Sl,Mp,.Step 3: From fuzzy nano topologies forτj and νk:(i)
τ1∗=0ℱ,1ℱ,0.9,0.3,0.2.(ii)
τ2∗=0ℱ,1ℱ,0.8,0.1,0.4,0.3.(iii)
τ3∗=0ℱ,1ℱ,0.8,0.3,0.2,0.4.(iv)
τ4∗=0ℱ,1ℱ,0.3,0.1,0.2,0.4,0.9.(v)
τ5∗=0ℱ,1ℱ,0.3,0.6,0.4,0.7.(i)
ν1∗=0ℱ,1ℱ,0.2,0.7,0.9.(ii)
ν2∗=0ℱ,1ℱ,0.9,0.2.(iii)
ν3∗=0ℱ,1ℱ,0.9,0.1,0.2.(iv)
ν4∗=0ℱ,1ℱ,0.6,0.1,0.2,0.9.(v)
ν5∗=0ℱ,1ℱ,0.1,0.9,0.4,0.3.Step 5: Find fuzzy score functions: (i)ℱSFτ1=0.48. (ii) ℱSFτ2=0.4333. (iii) ℱSFτ3=0.45. (iv) ℱSFτ4=0.4143. (v) ℱSFτ5=0.5.(i)ℱSFν1=0.56. (ii) ℱSFν2=0.525. (iii) ℱSFν3=0.44. (iv) ℱSFν4=0.4667. (v) ℱSFν5=0.45.Step 6: Final decision: Arrange fuzzy nano score values for the alternativesτ1, τ2, τ3, τ4, τ5 and the attributes ν1, ν2, ν3, ν4, ν5 in ascending order. We get the following sequences τ5≤τ1≤τ3≤τ2≤τ4 and ν1≤ν2≤ν4≤ν5≤ν3. 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.Table 1
Fuzzy values for patients.
Symptoms/PatientPat1Pat2Pat3Pat4Pat5Weight gain0.90.80.00.30.3Tiredness0.00.10.80.10.6Myalgia0.30.80.30.20.3Swelling of legs0.90.40.20.40.4Mensus Problem0.20.30.40.90.7Table 2
Fuzzy values for disease.
Symptoms/patientsWeight gainTirednessMyalgiaSwelling of legsMensus problemLymphedema0.00.20.70.90.2Insomnia0.00.90.20.20.2Hypothyroidism0.90.10.00.10.2Menarche0.60.10.20.20.9Arthritis0.00.10.90.40.3Figure 2
Fuzzy score values for diseases.Figure 3
Fuzzy Score values for Patients.


## 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 nanoZ 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.



---

*Source: 3364170-2022-05-11.xml*