Pythagorean Nanogeneralized Closed Sets with Application in Decision-Making

,


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 + ½ 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][9][10].Following this, intuitionistic topological spaces were developed into ideas as separation axioms, connectedness, and categorical property [11][12][13][14][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][26][27][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 decisionmaking 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][32][33][34][35].Our motivation for the work is that this is still a developing area in fuzzy mathematics, and we want to produce more theoretical concepts and show the application of the work in some real-life situations by combining it with wide-area decision-making.There are many existing models which are still developing in this particular area but when we deal with more fuzzified data, this method is more useful without reducing the constraint when compared to the other concepts.The proposed concepts and model have the more fuzzified values as information but still hold the same condition as the other models, which has a great advantage in dealing with the more vague details of the problem.
The following is how the article is organized: In Section 2, we define generalized closed sets of PNTS along with its characterizations.Sections 3 discusses the generalized semiclosed sets of PNTS.In Section 4, we present an MADM algorithm by using Pythagorean nanotopology, illustrate with the help of numerical example, and conclude in Section 5.

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.
Proof.Let T be a PNgC and F be PNC subset of PNclð TÞ − T: Then, T ⊆ F c , and Proof.Since H is PNgC, PNclðHÞ − H has no nonempty PNC subset.Since PNclðTÞ − T ⊆ PNclðHÞ − H, PNclð TÞ − T also does not have any nonempty PNC set.Therefore, T is PNgC.☐Theorem 10.Let U and V be two PNT spaces and H ⊆ V ⊆ U and H be PNgC in U: Then, H is PNgC in V: ðUÞ, and PNB R ðUÞ are the only sets which are PNO as well as PNC in U, when PNU R ðUÞ = U: Thus, The PN closure of each PNO set in U is PNO.U is extremally disconnected, and hence, PNU R ðUÞ = U: Proof.We know that every PNC 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 .☐ 3 Journal of Function Spaces Theorem 14. H is PNgO iff F ⊆ PNint ðHÞ whenever F is PNC and F ⊆ H: Conversely, let F ⊆ PNint ðHÞ for every PNC set such that F ⊆ H, and let G be PNO such that [ A new decision-making approach using PN topology and a methodological approach for selecting the right alternatives is proposed.

Algorithm
Step 1.Consider the universe D and attributes E.
Step 2. Make a fuzzy Pythagorean matrix of attributes versus objects.
Step 3. Define R on D to represent the indiscernibility relation.
Step 4. Build the Pythagorean fuzzy nanotopology τ: Step 5. Find the score values by using the score function 1/ k∑ k i=1 ½1/2f1 + m − n:mg (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.

Numerical Example.
The proposed algorithm helps to find the suitable choice among all the options (set of objects).We choose any random situation for this decision-making process.As in Algorithm, using the PNT S 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 = fd 1 , d 2 , d 3 , d 4 , d 5 g where d i ði = 1, 2, 3, 4, 5Þ.Consider the criteria for deciding to pick a hotel.The attributes are E = ft 1 , t 2 , t 3 , t 4 , t 5 g where t i , ði = 1, 2, 3, 4, 5Þ stands for criteria clean and tidy, good food, reasonable price, customer driven, and location, respectively.
Step 1.Let D = fd 1 , d 2 , d 3 , d 4 , d 5 g be the set of objects and E = ft 1 , t 2 , t 3 , t 4 , t 5 g be the set of attributes for the objects.
Step 2. In the matrix of Pythagorean fuzzy relationship between hotels, attributes are developed as in Table 1.
Step 3. The indiscernibility relation for the objects is constructed as Step 4. Build the PNTS for each hotel ðd i Þ with respect to the attributes. .Thus, the hotel with maximum value and in the first position is chosen as the optimal decision (i.e., d 4 ).

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 indetermi-nacy part.When compared to the other sets and models, this plays the upper hand.

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.
where PNcl H ðTÞ is the PN closure of T in H: Then, PNclðTÞ ⊆ H ∩ G and H ⊆ G S ½PNclðTÞ c : Also, G S ½PNclðTÞ c is PNO.Since H is PNgC in U,PNclðHÞ ⊆ G S ½PNclðTÞ c : Therefore, PNclðTÞ ⊆ PNclðHÞ ⊆ G S ½PNclðTÞ c , since T ⊆ H: Therefore, PNclðTÞ ⊆ G: Thus, for every PNO set G in U containing T, PNclðTÞ ⊆ G: Therefore, T is P NgC in U:☐ Corollary 8.If H is PNgC and and hence, H is PNgO in U:☐ Definition 15.If H and T are subset of PNTSU, then H and T are said to be PN separated (PNS), if H Proof. H c and T c are PNgC and hence H c S T Proof.H c and T c are PNS and PNgO, and hence, H c S T Theorem 16.If H and T are PNS and PNgO, then H S T is PNgO.Proof.Let F be PNC in U such that F ⊆ H S T: Since H and T are PNS, PNclðHÞ T T = ∅: Therefore, no element of PNclðHÞ belongs to T: Thus, no element of F T PNclðHÞ belongs to T: Hence, every element of F T PNc lðHÞ belongs to H, since F ⊆ H S T: That is, F T PNclð HÞ ⊆ H: Thus, F T PNclðHÞ is PNC subset of H: Since H is PNgO, F T PNclðHÞ ⊆ PNintðHÞ: Theorem 17.If H and T are PNgO in U, then H T T is PNgO.c = ðH T TÞ c is PNgC and hence ðH T TÞ is PNgO.☐Theorem 18.If H and T are PNgC sets such that H c and T c are PNS, then H T T is PNgC.c is PNgO.Therefore, H T T is PNgC.☐Theorem 19.H is PNgO iff G = U where G is PNO and PNintðHÞ.

Theorem 23 .
assume that whenever G is PNO and PN intðHÞ S H c ⊆ G, then G ⊆ U. Let F be PNC such that F ⊆ H.Then, PNintðHÞ S H c ⊆ PNintðHÞ S F c which is PNO.Therefore, PNintðHÞ S H c = U: That is, F ⊆ PNi ntðHÞ, since every x ∈ F, belongs to PNintðHÞ.Thus, F ⊆ PNintðHÞ whenever F is PNC and F ⊆ H: Therefore, H is PNgO.☐Theorem20.If PNintðHÞ ⊆ T ⊆ H and if H is PNgO, then T is also PNgO.Proof.H c ⊆ T c ⊆ PNclðH c Þ where H c is PNgC, and hence, T c is PNgC.Therefore, T is PNgO.☐Theorem21.H is PNgC if and only if PNclðHÞ − H is PNgO.Proof.Let H be PNgC.Let F be a PNC such that F ⊆ P NclðHÞ − H: Then, F = ∅, since PNclðHÞ − H cannot have any nonempty closed set.Therefore, F ⊆ PNint ðPNclðH Þ − HÞ, and hence, PNclðHÞ − H is PNgO.In a PNTS (U, τ R ðXÞ, if PNL R ðXÞ = PN U R ðXÞ, then any set H such that HUPNL R ðXÞ is the only PNgC set in U: Proof.When PNL R ðXÞ = 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, PNclðHÞ ⊆ G for every PNO set G having H: Thus, every subset H of U is PNg-C, if P NL R ðXÞ = PNU R ðXÞ = U.When PNL R ðXÞ ≠ U, the PN O sets in U are U, ∅, and PNL R ðXÞ.If H ⊆ PNL R ðXÞ, then PNO sets having H are PNL R ðXÞ and U. Also, PN L R ðXÞ ≠ U: And PNclðHÞUPNL R ðXÞ.Therefore, H is not PNgC.If HUPNL R ðXÞ, then U is the only PNO set holding H and hence PNclðHÞ ⊆ G for every PNO set G ⊇ H. Therefore, H is PNgC.Thus, only those sets H such that HUPNL R ðXÞ are PNgC, if PNL R ðXÞ = PNU R ðXÞ .☐IfPNLRðXÞ= ∅ and PNU R ðXÞ ≠ U in a P NTS, then those sets H for which HUPNU R ðXÞ are the only PNgC sets.Proof.τRðXÞ=f∅, U, PNU R ðXÞg: If H ⊆ PNU R ðXÞ, then U and PNU R ðXÞ are the PNO set containing H. P NclðHÞ = U; hence, PNclðHÞUPNU R ðXÞ.Thus, PNclð HÞUG when G = PNU R ðXÞ.Therefore, H is not PNgC.But, if HUPNU R ðXÞ, then U is the only PNO set that contains H and hence PNclðAÞ ⊆ G whenever G is PNO and G ⊇ H: Therefore, H is PNgC.Thus, only those subsets H of U such that HUPNU R ðXÞ are PNgC in U, if PN L R ðXÞ = ∅ and PNU R ðXÞ ≠ U:☐ Theorem 24.If PNL R ðXÞ ≠ ∅ and PNU R ðXÞ = U, then every subset H of U is PNgC.If PNL R ðXÞ ≠ PNU R ðXÞ and PNL R ðXÞ ≠ ∅ , PNU R ðXÞ ≠ U, and only those subsets H of U such that HUPNU R ðXÞ are PNgC in U: Proof.τRðXÞ=f∅, U, PNL R ðXÞ, PNU R ðXÞ, PNB R ðXÞg.If H ⊆ PNL R ðXÞ, then PNO sets containing H are PN L R ðXÞ, PNU R ðXÞ, and U.But PNclðHÞ = ½PNB R ðXÞ c = PNL R ðXÞ S ½PNU R ðXÞ c UPNL R ðXÞ, since ½PNU R ðXÞ c ≠ ∅.Therefore, H is not PNgC.If H ⊆ PN B R ðXÞ, then PNB R ðXÞ, PNU R ðXÞ, and U are the PNO sets containing H and PNclðHÞ = ½PNL R ðXÞ c = PNB R ð XÞ S ½PNU R ðXÞ c UPNB The PN semigeneralized interior of A, symbolized by PNsgintðHÞ, is defined as the largest PNsgO set in H: Remark 28.For subsets H and T of a PNTSðu, τ R ðXÞÞ, That is, ðPNsclðHÞ T H c Þ c ⊆ F c .Therefore, H S ðPNsintðHÞÞ c ⊆ F c .Thus, F c is PNSO, and H ⊆ F c : Since H is PNsgC, PNsclðHÞ ⊆ F c .That is, F ⊆ ðPNsclðHÞÞ c .Thus, F ⊆ ðPNsclðH:Þ T ðPNsclðHÞÞ c = ∅: Therefore, F = ∅: Conversely, let PNsclðHÞ − H have no nonempty, P NSC set.Let G be PNSO in U such that H ⊆ G.If PN sclðHÞUG, then PNsclðHÞ T G c ≠ ∅.And PNsclðHÞ T G c ⊆ PNsclðHÞ − H, since H ⊆ G. Thus, PNsclðHÞ T G c is a nonvoid PNSC subset of PNsclðHÞ − H, which is contradiction.Therefore, PNsclðHÞ ⊆ G whenever G is P NSO and H ⊆ G.That is, H is PNsgC in U:☐ Theorem 31.Let H be PNsgC.Then, H is PNSC iff P NsclðHÞ − H is PNSC.Proof.Let H be PNsgC.If H if PNSC, PNsclðHÞ = H and hence PNsclðHÞ − H = ∅ which is PNSC.Conversely, let PNsclðHÞ − H be PNSC.Then, PNsclðHÞ − H is PNsgC.Then, PNsclðHÞ − H does not contain any nonempty, PNSC set.Therefore, PNsclðHÞ − H = ∅ .That is, PNsclðHÞ = H.Therefore, H is PNSC.Now, we derive the forms of PN semigeneralized closed sets for various cases of approximations.☐Theorem32.If PNL R ðXÞ = PNU R ðXÞ in a PNTSU, then any H ⊆ ½PNL R ðXÞ c and ½PNL R ðXÞ c S T where T ⊆ PNL R ðXÞ are the only PNsgC sets in U: Proof.When PNL R ðXÞ = PNU R ðXÞ, τ R ðXÞ = f∅, U, PN L R ðXÞg: Also, ∅ and any H ⊆ PNL R ðXÞ are the only PN SO sets in U: If H ⊆ PNL R ðXÞ, then PNsclðHÞ = U and the PNSO sets containing H are those sets T for which PNL R ðXÞ ⊆ T. Thus, PNsclðHÞ ⊆ G, not for every PNS OG such that H ⊆ G. Therefore, H is not PNsgC.If H ⊆ ½PNL R ðXÞ c , then PNsclðHÞ − H, since any subset of ½PNL R ðXÞ c is PNS C in U: Thus, PNsclðHÞ = H ⊆ G whenever G is PNSO and H ⊆ G: Therefore, H is PNsgC.If PNL R ðXÞ ⊆ G and H ≠ U, PNsclðHÞ = U and the PNSO sets containing H are H and U. Therefore, PNsclðHÞUH.Therefore, any H ⊇ PNL R ðXÞ and H ≠ U are not PNsgC.If ½PNL R ðXÞ c ⊆ H, then PNsclðHÞ ⊆ G whenever G is PN SO and G ⊆ H, since U is the only PNSO set containing H. Therefore, if ½PNL R ðXÞ c ⊆ H, then H is PNsgC.When H has at least one element of PNL R ðXÞ and exactly one element of ½PNL R ðXÞ c where PNL R ðXÞ is not a singleton set, PNsclðHÞ = U: But union of that element and PNL R ðXÞ is a PNSO set containing H and PNsclðHÞ = UUPNL R ð XÞ union with that element.Therefore, H is not PNsgC.Thus, the only PNsgC sets in U are subsets of ½PNL R ðXÞ c and any H ⊃ ½PNL R ðXÞ c :☐ Theorem 33.If PNL R ðXÞ = ∅ and PNU R ðXÞ ≠ U, then the only PNsgC sets in U are subsets of ½PNU R ðXÞ c and any H ⊃ ½PNU R ðXÞ c : Proof.If PNL R ðXÞ = ∅ and PNU R ðXÞ = U, then τ R ðXÞ = f∅, U, PNU R ðXÞg.Also, ∅ and those sets H for which H ⊇ PNU R ðXÞ are the only PNSO sets in U: Therefore, the sets H for which H ⊆ ½PNU R ðXÞ c are the only PNS C sets in U: If H ⊆ PNU R ðXÞ, then PNsclðHÞ = U: But PNU R ðXÞ is a PNSO set containing H, for which PNsc lðHÞUG: Therefore, H is not PNsgC.If PNU R ðXÞ ⊆ H and H ≠ U, then PNsclðHÞ = U: But, for G = H, which is PNSO set containing itself, PNsclðHÞUG: Therefore, H is not PNsgC.If H ⊆ ½PNU R ðXÞ c , then PNsclðHÞ = H and hence for every PNSO set G such that H ⊆ G, PNsc lðHÞ ⊆ G: Therefore, H is PNsgC.If ½PNU R ðXÞ c ⊆ H, then U is the only PNSO set holding H and hence PNs clðHÞ ⊆ G whenever G is PNSO and H ⊆ G: Therefore, H is PNsgC.If H has one element of PNU R ðXÞ and at least one element of ðPNU R ðXÞÞ c , then PNsclðHÞ = U: Since any set having PNU R ðXÞ is PNSO in U, H S ðPNU R ðXÞÞ and any set having H S ðPNU R ðXÞÞ are PNSO sets containing H: But, PNsclðHÞ = UUH S ðPNU R ðXÞÞ: Therefore, H is not PNsgC in U: Thus, only subsets of ðPNU R ðXÞÞ c and any H ⊃ ½PNU R ðXÞ c are P NsgC in U when PNL R ðXÞ = ∅ and PNU R ðXÞ ≠ U:☐ Theorem 34.If PNU R ðXÞ = U and PNL R ðXÞ ≠ ∅ in a P NTSU, then every subset of U is PNsgC.Proof.∅,U,PNLRðXÞ, and PNB R ðXÞ are the only sets in U which are PNO, PNSO, and PNSC in U: If H ⊆ P NL R ðXÞ, then PNL R ðXÞ and U are the only PNSO sets containing H and PNsclðHÞ = PNL R ðXÞ: Therefore, PN sclðHÞ ⊆ G whenever G is PNSO and H ⊆ G: Thus, H is PNsgC.If H ⊆ PNB R ðXÞ, then PNB R ðXÞ and U are the only PNSO sets containing H and PNsclðHÞ = PNB R ð XÞ: Therefore, H is PNsg-C.If PNL R ðXÞ ⊂ H or PNB R ð XÞ ⊂ H, then U is the only PNSO set containing H and hence H is PNsgC.If H contains atleast one element of PNL R ðXÞ and at least one element of PNB R ðXÞ, then U is the only PNSO set containing H: Therefore, H is PN sgC.Thus, every subset of U is PNsgC, if PNU R ðXÞ = U and PNL R ðXÞ ≠ ∅:☐ Definition 35.Let ðU, τ R ðXÞÞ and ðV, τ R ′ ðYÞÞ be two PNTSs.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 V Theorem 36.Every PN continuous (PNCN) function is PNsgCN.Proof.If f : U ⟶ V is PNCN on U and if G is PNO in V, then f −1 ðGÞ is PNO in U: Therefore, f −1 ðGÞ is PNsgO,since any PNO set is PNSO and any PNSO is PNSgO.MADM is a method for selecting the best solution with the highest level of satisfaction from a set of alternatives.Multiple attributes are used to represent these types of MADM problems, which occur in most real-time situations.When it comes to dealing with real-life problems, collecting vague details is done with the help of attributes for the particular object and the decision-making technique is applied for the list of objects considered.Many models already exist for the decision-making problems, but the proposed algorithm deals with membership and nonmembership which has more advantage in fuzziness than intuitionistic fuzzy set and fuzzy set theory.Many types of models exist for the different developments of topological spaces, but for the different category of topological spaces, this method is proposed.The proposed algorithm describes how PN topology influences decision-making.
Conversely, if PNclðHÞ − H is PNgO and G is PNO such that H ⊆ G, then PNclðHÞ T G c ⊆ PNclðHÞ T H c = PNclðHÞ − H where PNclðHÞ T G c is PNC.Since PNc lðHÞ − H is PNgO, PNclðHÞ T G c ⊆ PNint½PNclðHÞ − H = ∅: Therefore, PNclðHÞ T G c = ∅,and hence, PN clðHÞ ⊆ G: Thus, whenever G is PNO and H ⊆ G, PNclð HÞ ⊆ G:H is PNgC.☐Theorem 22. R ðXÞ: Therefore, H is not PNg C. If H ⊆ PNU R ðXÞ, neither a subset of PNL R ðXÞ is a P NO set containing H for which PNclðHÞUPNU R ðXÞ.Therefore, H is not PNgC.If HUPNU R ðXÞ, then U is the only PNO set containing H and hence PNclðHÞ ⊆ G for every PNOG ⊇ H. Therefore, H is PNgC.Thus, only those H ⊆ U for which HUPNU R ðXÞ are PNgC in U.☐ 3. Pythagorean Nanosemigeneralized Closed Sets As we have defined in the last section, we have extended the concept of PN generalized closed sets to PN semigeneralized closed sets and investigated their properties.Definition 26.A subset H of PNTSU is said to be PN semigeneralized closed (PNsgC), if PNsclðHÞ ⊆ G whenever G is PNSO and H ⊆ G: The set H is named as PN semigeneralized open (PNsgO) if H c is PNsgC.Definition 27.If H ⊆ U, then the PN semigeneralized closure represented by PNsgclðHÞ is defined as the smallest PNsgC set having H: Theorem 30.A set H is PNsgC in U iff PNsclðHÞ − H has no nonempty, PNSC set.Proof.Let H be a PNsgC and F be a PNSC subset of P NsclðHÞ − H: Then, ðPNsclðHÞ − HÞ c ⊆ F c , and F c is P NSO.

Table 1 :
Pythagorean fuzzy system of relationship between hotels and attributes.