# Pythagorean Nanogeneralized Closed Sets with Application in Decision-Making

Authors: D. Ajay; J. Joseline Charisma; T. Petaratip; P. Hammachukiattikul; N. Boonsatit; Jihad A. Younis; G. Rajchakit
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 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 −0,1+ 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-α-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 Section2, 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

PNTS has been defined in [28], the weak forms of open sets of PNTS have been defined, and their properties were investigated in [26, 28]. In this section, as an extension of these ideas of PNTS, the generalized closed sets have been developed and various characterizations of these sets have been examined. Throughout this paper, Pythagorean nano is denoted by PN.Definition 1 (see [36]).
Let the Universe beU and equivalence relation on U be R, and if τRY=∅,U,PNLRY,PNURY,PNBRY where Y⊆U, τRY holds the following axioms:
(1)
∅,U∈τRY(2)
IfTk∈τRY fork=1,2,⋯, then ⋃k=1∞Tk∈τRY(3)
IfTk∈τRY for k=1,2,⋯,n, then ⋂k=1nTk∈τRY
Then,τRY is termed as PN topology (PNT) on Uw.r.tY whereas ∅=w,0,1∀w∈U,U=w,1,0∀w∈U,PNLRY=w,θLYw,ωLYwz∈wR,w∈U, PNURY=w,θRYw,ωRYwz∈wR,w∈U, and PNBRY=PNURY−PNLRY where θLYw=∧k∈wRθYk,ωLYw=∨y∈wRωLYk and θRYw=∨k∈wRθYk,ωRYw=∧k∈wRωYk.
We callU,τRY as Pythagorean nanotopological spaces (PNTS). The elements of τRY are called PN open (PNO) sets.Definition 2.
A subsetT of PNTSU,τRY is PN generalized closed (PNgC) if PNclT⊆G whenever T⊆G, G is PNO in U.Theorem 3.
T is PNgC in a PNTU iff PNclT−T has no nonvoid PNC set.Proof.
LetT be a PNgC and F be PNC subset of PNclT−T. Then, T⊆Fc, and Fc is PNO. Since T is PNgC, PNclT⊆Fc. Therefore, F⊆PNclTc and F⊆PNclT∩PNclTc=∅. That is, F=∅; thus, PNclT−T cannot have any nonempty PNC set.
Conversely, letPNclT−T has no nonvoid PNC set. G be a PNO such that T⊆G. Then, PNclT∩Gc is a PNC subset of PNclT−T, since PNclT∩Gc≤PNclT∩Tc, as T⊆G. Therefore, PNclT∩Gc=∅. Since PNclT−T does not have any nonempty PNC set. Hence, PNclT⊆G. Therefore, T is PNgC in U.☐Theorem 4.
EveryPNC set is PNgC.Proof.
IfT is PNC, then PNclT=T. Therefore, for every PNO set G such that T⊆G,PNclT⊆G and hence T is PNgC.☐Theorem 5.
APNgC set H is PNC in U if and only if PNclH−H is PNC.Proof.
IfH is PNC in U, then PNclH=H and hence PNclH−H=∅ which is PNC. Conversely, PNclH−H is PNC, and PNclH−H=∅. Therefore, PNclH=H, and hence, H is PNC.☐Theorem 6.
IfH and T are PNgC, then H⋃T is PNgC.Proof.
LetG be PNO in U such that H⋃T⊆G. Then, Hand T⊆G, since H is PNgC and G is a PNO set having PNclH⊆G.
Similarly,PNclT⊆G.PNclH⋃T=PNclH⋃PNclT⊆G.H⋃T is PNgC.☐Theorem 7.
LetH be a PNgC subset of U and T⊆H be a PNgC set to H. Then, T is PNgC to U.Proof.
LetG be a PNO set in U such that T⊆G. Then, T⊆H∩G.H∩G is a PNO to H containing T. Since T is PNgC to H,PNclHT=H∩G, where PNclHT is the PN closure of T in H. Then, PNclT⊆H∩G and H⊆G⋃PNclTc. Also, G⋃PNclTc is PNO. Since H is PNgC in U,PNclH⊆G⋃PNclTc. Therefore, PNclT⊆PNclH⊆G⋃PNclTc, since T⊆H. Therefore, PNclT⊆G. Thus, for every PNO set G in U containing T, PNclT⊆G. Therefore, T is PNgC in U.☐Now, we prove the intersection of aPNgC and a PNC is PNgC.Corollary 8.
IfH is PNgC and F is PNC in U, then H∩F is PNgC.Proof.
SinceH∩F is PNC to H, it is PNgC to H. Since H∩F⊆H where H is PNgC in U and H∩F is PNgC to H, H∩F is PNgC in U by the above theorem.☐Theorem 9.
IfH is PNgC and H⊆T⊆PNclH, then T is PNgC.Proof.
SinceH is PNgC, PNclH−H has no nonempty PNC subset. Since PNclT−T⊆PNclH−H, PNclT−T also does not have any nonempty PNC set. Therefore, T is PNgC.☐Theorem 10.
LetU and V be two PNT spaces and H⊆V⊆U and H be PNgC in U. Then, H is PNgC in V.Proof.
LetG1 be PNO in V such that H⊆G1. Then, G1=V∩G where G is PNO in U and H=V∩G. Therefore, H⊆G. That is, G is PNO containing H. Since H is PNgC in U,PNclH⊆G. Therefore, V∩PNclH⊆V∩G. That is, PNclH⊆G1, for every PNOG1 in V such that G1⊇H. Therefore, H is PNgC in V.☐Theorem 11.
Every subset ofPNTU is PNgC iff PNURU=U.Proof.
LetPNURU=U and H⊆U. Let G be PNO in U such that H⊆G. Then, PNclH⊆PNclG⊆G, since U, ∅, PNLRU, and PNBRU are the only sets which are PNO as well as PNC in U, when PNURU=U. Thus, PNclH⊆G whenever G is PNO and H⊆G.H is PNgC, if PNURU=U.☐Conversely, assume that every subset ofU is PNgC. Let G∈τRU. Then, G is PNgC. Since G is PNO and G⊆G, PNclG⊆G; hence, G is PNC. Thus, whenever G is PNO in U, PNclG⊆G and PNclG is PNO in U. The PN closure of each PNO set in U is PNO. U is extremally disconnected, and hence, PNURU=U.Definition 12.
A setH in a PNTSU is PN generalized open (PNgO) if Hc is PNgC.Theorem 13.
EveryPNO set is PNgO.Proof.
We know that everyPNC is PNgC. Thus, if we take complement, we get every PNO is PNgO. But the converse need not to be true. That is, every PNgO need not be PNO.☐Theorem 14.
H is PNgO iff F⊆PNintH whenever F is PNC and F⊆H.Proof.
LetH be a PNgO in U. Then, Hc is PNgC. Therefore, PNclHc⊆G whenever G is PNO and Hc⊆G. Let F be PNC and F⊆H. Then, Hc⊆Fc, and Fc is PNO. Therefore, PNclHc⊆Fc. That is, U−PNintH⊆U−F. Hence, F⊆PNintH.
Conversely, letF⊆PNintH for every PNC set such that F⊆H, and let G be PNO such that Hc⊆G. Then, Gc⊆H where Gc is PNC. Therefore, Gc⊆PNintH. Therefore, U−PNintH⊆G. Therefore, PNclU−H⊆G. That is, PNclHc⊆G. Therefore, Hc is PNgC, and hence, H is PNgO in U.☐Definition 15.
IfH and T are subset of PNTSU, then H and T are said to be PN separated (PNS), if H⋂PNclT=∅ and T⋂PNclH=∅.Theorem 16.
IfH and T are PNS and PNgO, then H⋃T is PNgO.Proof.
LetF be PNC in U such that F⊆H⋃T. Since H and T are PNS, PNclH⋂T=∅. Therefore, no element of PNclH belongs to T. Thus, no element of F⋂PNclH belongs to T. Hence, every element of F⋂PNclH belongs to H, since F⊆H⋃T. That is, F⋂PNclH⊆H. Thus, F⋂PNclH is PNC subset of H. Since H is PNgO, F⋂PNclH⊆PNintH.
Similarly,F⋂PNclT⊆PNintT. Since F⊆H⋃H,F=F⋂H⋃T=F⋂H⋃F⋂T⊆F⋂PNclH⋃F⋂PNclT⊆PNintH⋃PNintT⊆PNintH⋃T. Thus, F⊆PNintH⋃T, whenever F is PNC and F⊆H⋃T.☐Therefore,H⋃T is PNgO.Theorem 17.
IfH and T are PNgO in U, then H⋂T is PNgO.Proof.
Hc and Tc are PNgC and hence Hc⋃Tc=H⋂Tc is PNgC and hence H⋂T is PNgO.☐Theorem 18.
IfH and T are PNgC sets such that Hc and Tc are PNS, then H⋂T is PNgC.Proof.
Hc and Tc are PNS and PNgO, and hence, Hc⋃Tc is PNgO. Therefore, H⋂T is PNgC.☐Theorem 19.
H is PNgO iff G=U where G is PNO and PNintH.
(1)⋃Hc⊆G.Proof.
LetH be PNgO. Let G be PNO such that PNintH⋃Hc⊆G. Then, Gc⊆H⋂PNclHc⊆PNclHc−Hc. Since Hc is PNgC, PNclHc−Hc cannot have any nonempty PNC set. But Gc is PNC subset of PNclHc−Hc. Therefore, Gc⊆∅. That is, G=U.
Conversely, assume that wheneverG is PNO and PNintH⋃Hc⊆G, then G⊆U. Let F be PNC such that F⊆H. Then, PNintH⋃Hc⊆PNintH⋃Fc which is PNO. Therefore, PNintH⋃Hc=U. That is, F⊆PNintH, since every x∈F, belongs to PNintH. Thus, F⊆PNintH whenever F is PNC and F⊆H. Therefore, H is PNgO.☐Theorem 20.
IfPNintH⊆T⊆H and if H is PNgO, then T is also PNgO.Proof.
Hc⊆Tc⊆PNclHc where Hc is PNgC, and hence, Tc is PNgC. Therefore, T is PNgO.☐Theorem 21.
H is PNgC if and only if PNclH−H is PNgO.Proof.
LetH be PNgC. Let F be a PNC such that F⊆PNclH−H. Then, F=∅, since PNclH−H cannot have any nonempty closed set. Therefore, F⊆PNintPNclH−H, and hence, PNclH−H is PNgO.
Conversely, ifPNclH−H is PNgO and G is PNO such that H⊆G, then PNclH⋂Gc⊆PNclH⋂Hc=PNclH−H where PNclH⋂Gc is PNC. Since PNclH−H is PNgO, PNclH⋂Gc⊆PNintPNclH−H=∅. Therefore, PNclH⋂Gc=∅, and hence, PNclH⊆G. Thus, whenever G is PNO and H⊆G, PNclH⊆G.H is PNgC.☐Theorem 22.
In aPNTS (U,τRX, if PNLRX=PNURX, then any set H such that HÚPNLRX is the only PNgC set in U.Proof.
WhenPNLRX=U, U and ∅ are the only PNO sets and hence for any subset H of U, U is the only PNO set holding it. Therefore, PNclH⊆G for every PNO set G having H. Thus, every subset H of U is PNg-C, if PNLRX=PNURX=U. When PNLRX≠U, the PNO sets in U are U, ∅, and PNLRX. If H⊆PNLRX, then PNO sets having H are PNLRX and U. Also, PNLRX≠U. And PNclHÚPNLRX. Therefore, H is not PNgC. If HÚPNLRX, then U is the only PNO set holding H and hence PNclH⊆G for every PNO set G⊇H. Therefore, H is PNgC. Thus, only those sets H such that HÚPNLRX are PNgC, if PNLRX=PNURX.☐Theorem 23.
IfPNLRX=∅ and PNURX≠U in a PNTS, then those sets H for which HÚPNURX are the only PNgC sets.Proof.
τRX=∅,U,PNURX. If H⊆PNURX, then U and PNURX are the PNO set containing H. PNclH=U; hence, PNclHÚPNURX. Thus, PNclHÚG when G=PNURX. Therefore, H is not PNgC. But, if HÚPNURX, then U is the only PNO set that contains H and hence PNclA⊆G whenever G is PNO and G⊇H. Therefore, H is PNgC. Thus, only those subsets H of U such that HÚPNURX are PNgC in U, if PNLRX=∅ and PNURX≠U.☐Theorem 24.
IfPNLRX≠∅ and PNURX=U, then every subset H of U is PNgC.Theorem 25.
IfPNLRX≠PNURX and PNLRX≠∅, PNURX≠U, and only those subsets H of U such that HÚPNURX are PNgC in U.Proof.
τRX=∅,U,PNLRX,PNURX,PNBRX. If H⊆PNLRX, then PNO sets containing H are PNLRX, PNURX, and U. But PNclH=PNBRXc=PNLRX⋃PNURXcÚPNLRX, since PNURXc≠∅. Therefore, H is not PNgC. If H⊆PNBRX, then PNBRX, PNURX, and U are the PNO sets containing H and PNclH=PNLRXc=PNBRX⋃PNURXcÚPNBRX. Therefore, H is not PNgC. If H⊆PNURX, neither a subset of PNLRX is a PNO set containing H for which PNclHÚPNURX. Therefore, H is not PNgC. If HÚPNURX, then U is the only PNO set containing H and hence PNclH⊆G for every PNOG⊇H. Therefore, H is PNgC. Thus, only those H⊆U for which HÚPNURX are PNgC in U.☐


## 3. Pythagorean Nanosemigeneralized Closed Sets

As we have defined in the last section, we have extended the concept ofPN generalized closed sets to PN semigeneralized closed sets and investigated their properties.Definition 26.
A subsetH of PNTSU is said to be PN semigeneralized closed (PNsgC), if PNsclH⊆G whenever G is PNSO and H⊆G. The set H is named as PN semigeneralized open (PNsgO) if Hc is PNsgC.Definition 27.
IfH⊆U, then the PN semigeneralized closure represented by PNsgclH is defined as the smallest PNsgC set having H.
ThePN semigeneralized interior of A, symbolized by PNsgintH, is defined as the largest PNsgO set in H.Remark 28.
For subsetsH and T of a PNTSu,τRX,
(1)
U−PNsgintH=PNsgclU−H(2)
U−PNsgclH=PNsgintU−HTheorem 29.
EveryPNSC set is PNsgC.Theorem 30.
A setH is PNsgC in U iff PNsclH−H has no nonempty, PNSC set.Proof.
LetH be a PNsgC and F be a PNSC subset of PNsclH−H. Then, PNsclH−Hc⊆Fc, and Fc is PNSO. That is, PNsclH⋂Hcc⊆Fc. Therefore, H⋃PNsintHc⊆Fc. Thus, Fc is PNSO, and H⊆Fc. Since H is PNsgC, PNsclH⊆Fc. That is, F⊆PNsclHc. Thus, F⊆PNsclH.⋂PNsclHc=∅. Therefore, F=∅.
Conversely, letPNsclH−H have no nonempty, PNSC set. Let G be PNSO in U such that H⊆G. If PNsclHÚG, then PNsclH⋂Gc≠∅. And PNsclH⋂Gc⊆PNsclH−H, since H⊆G. Thus, PNsclH⋂Gc is a nonvoid PNSC subset of PNsclH−H, which is contradiction. Therefore, PNsclH⊆G whenever G is PNSO and H⊆G. That is, H is PNsgC in U.☐Theorem 31.
LetH be PNsgC. Then, H is PNSC iff PNsclH−H is PNSC.Proof.
LetH be PNsgC. If H if PNSC, PNsclH=H and hence PNsclH−H=∅ which is PNSC. Conversely, let PNsclH−H be PNSC. Then, PNsclH−H is PNsgC. Then, PNsclH−H does not contain any nonempty, PNSC set. Therefore, PNsclH−H=∅. That is, PNsclH=H. Therefore, H is PNSC.
Now, we derive the forms ofPN semigeneralized closed sets for various cases of approximations.☐Theorem 32.
IfPNLRX=PNURX in a PNTSU, then any H⊆PNLRXc and PNLRXc⋃T where T⊆PNLRX are the only PNsgC sets in U.Proof.
WhenPNLRX=PNURX, τRX=∅,U,PNLRX. Also, ∅ and any H⊆PNLRX are the only PNSO sets in U. If H⊆PNLRX, then PNsclH=U and the PNSO sets containing H are those sets T for which PNLRX⊆T. Thus, PNsclH⊆G, not for every PNSOG such that H⊆G.
Therefore,H is not PNsgC. If H⊆PNLRXc, then PNsclH−H, since any subset of PNLRXc is PNSC in U. Thus, PNsclH=H⊆G whenever G is PNSO and H⊆G. Therefore, H is PNsgC. If PNLRX⊆G and H≠U, PNsclH=U and the PNSO sets containing H are H and U. Therefore, PNsclHÚH. Therefore, any H⊇PNLRX and H≠U are not PNsgC. If PNLRXc⊆H, then PNsclH⊆G whenever G is PNSO and G⊆H, since U is the only PNSO set containing H. Therefore, if PNLRXc⊆H, then H is PNsgC. When H has at least one element of PNLRX and exactly one element of PNLRXc where PNLRX is not a singleton set, PNsclH=U. But union of that element and PNLRX is a PNSO set containing H and PNsclH=UÚPNLRX union with that element. Therefore, H is not PNsgC. Thus, the only PNsgC sets in U are subsets of PNLRXc and any H⊃PNLRXc.☐Theorem 33.
IfPNLRX=∅ and PNURX≠U, then the only PNsgC sets in U are subsets of PNURXc and any H⊃PNURXc.Proof.
IfPNLRX=∅ and PNURX=U, then τRX=∅,U,PNURX. Also, ∅ and those sets H for which H⊇PNURX are the only PNSO sets in U. Therefore, the sets H for which H⊆PNURXc are the only PNSC sets in U. If H⊆PNURX, then PNsclH=U. But PNURX is a PNSO set containing H, for which PNsclHÚG. Therefore, H is not PNsgC. If PNURX⊆H and H≠U, then PNsclH=U. But, for G=H, which is PNSO set containing itself, PNsclHÚG. Therefore, H is not PNsgC. If H⊆PNURXc, then PNsclH=H and hence for every PNSO set G such that H⊆G, PNsclH⊆G. Therefore, H is PNsgC. If PNURXc⊆H, then U is the only PNSO set holding H and hence PNsclH⊆G whenever G is PNSO and H⊆G. Therefore, H is PNsgC. If H has one element of PNURX and at least one element of PNURXc, then PNsclH=U. Since any set having PNURX is PNSO in U,
H⋃PNURX and any set having H⋃PNURX are PNSO sets containing H. But, PNsclH=UÚH⋃PNURX. Therefore, H is not PNsgC in U. Thus, only subsets of PNURXc and any H⊃PNURXc are PNsgC in U when PNLRX=∅ and PNURX≠U.☐Theorem 34.
IfPNURX=U and PNLRX≠∅ in a PNTSU, then every subset of U is PNsgC.Proof.
∅, U, PNLRX, and PNBRX are the only sets in U which are PNO, PNSO, and PNSC in U. If H⊆PNLRX, then PNLRX and U are the only PNSO sets containing H and PNsclH=PNLRX. Therefore, PNsclH⊆G whenever G is PNSO and H⊆G. Thus, H is PNsgC. If H⊆PNBRX, then PNBRX and U are the only PNSO sets containing H and PNsclH=PNBRX. Therefore, H is PNsg-C. If PNLRX⊂H or PNBRX⊂H, then U is the only PNSO set containing H and hence H is PNsgC. If H contains atleast one element of PNLRX and at least one element of PNBRX, then U is the only PNSO set containing H. Therefore, H is PNsgC. Thus, every subset of U is PNsgC, if PNURX=U and PNLRX≠∅.☐Definition 35.
LetU,τRX and V,τR′Y be two PNTSs. Then, a function f:U⟶V is named as
(1)
PN generalized continuous (PNgCN), if the inverse image of every PNC set in V is PNgC in U(2)
PN semigeneralized continuous (PNsgCN), if the inverse image of every PNC set in V is PNsgC in U(3)
PN semigeneralized closed (PNsgC) if the image of every PNC set in U is PNsgC in V(4)
PN semigeneralized open (PNsgO) if the image of every PNO set in U is PNsgO in VTheorem 36.
EveryPN continuous (PNCN) function is PNsgCN.Proof.
Iff:U⟶V is PNCN on U and if G is PNO in V, then f−1G is PNO in U. Therefore, f−1G is PNsgO, since any PNO set is PNSO and any PNSO is PNSgO. Therefore, f is PNsgCN.☐


## 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 howPN topology influences decision-making.A new decision-making approach usingPN topology and a methodological approach for selecting the right alternatives is proposed.


### 4.1. Algorithm

Step 1.
Consider the universeD and attributes E.Step 2.
Make a fuzzy Pythagorean matrix of attributes versus objects.Step 3.
DefineR on D to represent the indiscernibility relation.Step 4.
Build the Pythagorean fuzzy nanotopologyτ.Step 5.
Find the score values by using the score function1/k∑i=1k1/21+m−n.m (where m means membership, n.m means nonmembership, and k 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 thePNTS 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, D=d1,d2,d3,d4,d5 where dii=1,2,3,4,5. Consider the criteria for deciding to pick a hotel. The attributes are E=t1,t2,t3,t4,t5 where ti,i=1,2,3,4,5 stands for criteria clean and tidy, good food, reasonable price, customer driven, and location, respectively.Step 1.
LetD=d1,d2,d3,d4,d5 be the set of objects and E=t1,t2,t3,t4,t5 be the set of attributes for the objects.Step 2.
In the matrix of Pythagorean fuzzy relationship between hotels, attributes are developed as in Table1.Step 3.
The indiscernibility relation for the objects is constructed as(2)R=t1,t2,t3,t4,t5.Step 4.
Build thePNTS for each hotel di with respect to the attributes.
(3)τd1=ϕ,D,<.6,.4>,<.3,.3>,<.7,.3>,<.5,.1>,<.4,.6>,<.5,.7>,τd2=ϕ,D,<.5,.6>,<.5,.5>,<.6,.3>,<.7,.2>,<.5,.4>,τd3=ϕ,D,<.3,.5>,<.5,.7>,<.6,.4>,<.8,.4>,<.6,.5>,<.5,.4>,τd4=ϕ,D,<.3,.5>,<.4,.5>,<.7,.4>,<.9,.2>,<.7,.5>,<.6,.5>,τd5=ϕ,D,<.7,.7>,<.3,.7>,<.7,.6>,<.5,.3>,<.3,.6>.Step 5.
Computation of Pythagorean fuzzy score functions for the hotelsdi as in algorithm are as follows:
Score values of hotelsdii=1,2,⋯5 are
(4)Sd1=.5375,Sd2=.557,Sd3=.525,Sd4=.5625,Sd5=.471.Step 6.
Organizing the score values, we get the sequence of the hotels asd4>d2>d1>d3>d5. Thus, the hotel with maximum value and in the first position is chosen as the optimal decision (i.e., d4).Table 1
Pythagorean fuzzy system of relationship between hotels and attributes.
Rt1t2t3t4t5d1<.7,.4><.6,.3><.3,.2><.5,.1><.4,.3>d2<.5,.3><.6,.6><.7,.2><.6,.5><.5,.2>d3<.6,.4><.3,.5><.8,.5><.5,.4><.6,.7>d4<.3,.4><.7,.5><.5,.5><.9,.2><.4,.5>d5<.7,.6><.7,.7><.3,.3><.4,.3><.5,.7>


### 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.


## 4.1. Algorithm

Step 1.
Consider the universeD and attributes E.Step 2.
Make a fuzzy Pythagorean matrix of attributes versus objects.Step 3.
DefineR on D to represent the indiscernibility relation.Step 4.
Build the Pythagorean fuzzy nanotopologyτ.Step 5.
Find the score values by using the score function1/k∑i=1k1/21+m−n.m (where m means membership, n.m means nonmembership, and k 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 thePNTS 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, D=d1,d2,d3,d4,d5 where dii=1,2,3,4,5. Consider the criteria for deciding to pick a hotel. The attributes are E=t1,t2,t3,t4,t5 where ti,i=1,2,3,4,5 stands for criteria clean and tidy, good food, reasonable price, customer driven, and location, respectively.Step 1.
LetD=d1,d2,d3,d4,d5 be the set of objects and E=t1,t2,t3,t4,t5 be the set of attributes for the objects.Step 2.
In the matrix of Pythagorean fuzzy relationship between hotels, attributes are developed as in Table1.Step 3.
The indiscernibility relation for the objects is constructed as(2)R=t1,t2,t3,t4,t5.Step 4.
Build thePNTS for each hotel di with respect to the attributes.
(3)τd1=ϕ,D,<.6,.4>,<.3,.3>,<.7,.3>,<.5,.1>,<.4,.6>,<.5,.7>,τd2=ϕ,D,<.5,.6>,<.5,.5>,<.6,.3>,<.7,.2>,<.5,.4>,τd3=ϕ,D,<.3,.5>,<.5,.7>,<.6,.4>,<.8,.4>,<.6,.5>,<.5,.4>,τd4=ϕ,D,<.3,.5>,<.4,.5>,<.7,.4>,<.9,.2>,<.7,.5>,<.6,.5>,τd5=ϕ,D,<.7,.7>,<.3,.7>,<.7,.6>,<.5,.3>,<.3,.6>.Step 5.
Computation of Pythagorean fuzzy score functions for the hotelsdi as in algorithm are as follows:
Score values of hotelsdii=1,2,⋯5 are
(4)Sd1=.5375,Sd2=.557,Sd3=.525,Sd4=.5625,Sd5=.471.Step 6.
Organizing the score values, we get the sequence of the hotels asd4>d2>d1>d3>d5. Thus, the hotel with maximum value and in the first position is chosen as the optimal decision (i.e., d4).Table 1
Pythagorean fuzzy system of relationship between hotels and attributes.
Rt1t2t3t4t5d1<.7,.4><.6,.3><.3,.2><.5,.1><.4,.3>d2<.5,.3><.6,.6><.7,.2><.6,.5><.5,.2>d3<.6,.4><.3,.5><.8,.5><.5,.4><.6,.7>d4<.3,.4><.7,.5><.5,.5><.9,.2><.4,.5>d5<.7,.6><.7,.7><.3,.3><.4,.3><.5,.7>


## 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

PNTS 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 PNTS has been proposed and illustrated using a numerical example. Further, the proposed concept can be extended to strong open sets in PNTS and applied to real-life problems.



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*Source: 2571301-2021-12-26.xml*