Some GIS Topological Concepts via Neutrosophic Crisp Set Theory

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1 New Trends in Neutrosophi Theory and Appliations A.A.SALAMA, I.M.HANAFY, HEWAYDA ELGHAWALBY 3, M.S.DABASH 4,2,4 Department of Mathematis and Computer Siene, Faulty of Sienes, Port Said University, Egypt. s: 3 Faulty of Engineering, port-said University, Egypt. Some GIS Topologial Conepts via Neutrosophi Crisp Set Theory Abstrat In this paper we introdue and study the neutrosophi risp pre-open, semi-open, β- open set, neutrosophi risp ontinuity and neutrosophi risp ompat spaes are introdued. Furthermore, we investigate some of their properties and haraterizations. Possible appliation to GIS topology rules are touhed upon. Keywords Neutrosophi risp topologial spaes, neutrosophi risp sets, neutrosophi risp ontinuity, neutrosophi risp ompat spae.. Introdution Smarandahe [26, 27] introdued the notion of neutrosophi sets, whih is a generalization of Zadeh's fuzzy set [28]. In Zadah's sense, there is no preise definition for the set. Later on, Atanassov presented the idea of the intuitionisti fuzzy set [], where he goes beyond the degree of membership introduing the degree of non-membership of some element in the set. The new presented onepts attrated several authors to develop the lassial mathematis. For instane, Chang [2] and Lowen [6] started the disipline known as "Fuzzy Topology", where they forwarded the onepts from fuzzy sets to the lassial topologial spaes. Furthermore, Salama et al. [4, 7, 20] established several notations for what they alled, "Neutrosophi topologial spaes"]. In this paper, we study in more details some weaker and stronger strutures onstruted from the neutrosophi risp topology introdued in [7], as well as the onepts neutrosophi risp interior and the neutrosophi losure. The remaining of this paper is strutured as follows: in 2, some basi definitions are presented, while the new onepts of neutrosophi risp nearly open sets are introdued in 3, in addition to providing a study of some of its properties. The neutrosophi risp ontinuous funtion and neutrosophi risp ompat spaes are presented in 4 and 5, respetively. 35

2 Florentin Smarandahe, Surapati Pramanik (Editors) 2. Terminologies We reollet some relevant basi preliminaries, in partiular, the work introdued by We reollet some relevant basi preliminaries, in partiular, the work introdued by Smarandahe and Salama [7], Salama et al. [8] and Smarandahe [25,26,27]. The neutrosophi omponents T, I, F: X ]0, + [to represent the membership, indeterminay, and non-membership values of some universe X, respetively, where ]0, + [is the nonstandard unit Interval. Definition 2. [7] Let X be a non-empty fixed sample spae. A neutrosophi risp set (NCS for short) A is an objet having the form A = (A, A2, A3) where A, A2 and A3 are subsets of X. Where A ontains all those members of the spae X that aept the event A and A3 ontains all those members of the spae X that rejeted the event A, while A2 ontains those who stand in a distane from aepting or rejeting A. Definition 2.2 Salama [7] defined the objet having the form A = (A, A2, A3) to be ) (Neutrosophi Crisp Set with Type ),if satisfying A A2=, A A3 = and A2 A3 =. (NCS -Type ). 2) (Neutrosophi Crisp Set with Type 2), if satisfying A A2=, A A3 = and A2 A3 = and A A2 A3 = X (NCS -Type 2 ). 3) (Neutrosophi Crisp Set with Type 3) if satisfying A A2 A3 = and A A2 A3 = X. (NCS -Type3 for short). Every neutrosophi risp set A of a non-empty set X is obviously ancs having the form A = (A, A2, A3). Definition 2.3 [7] Let A = (A, A2, A3) a NCS on X, then the omplement of the set A, (A for short ) was presented in [7], to have one of the following forms: (C) A = (A, A 2, A 3 ) or (C2) A = (A3, A2, A) or (C3) A = (A3, A 2, A). Several relations and operations between NCS were defined in [7], whih we are introduing in the following: Definition 2.4 [7] Let X be a non-empty set, and NCSA and B in the form A = (A, A2, A3), B= (B, B2, B3), then we may onsider two possible definitions for subsets (A B). The onept of (A B) may be defined as two types: Type. A B A B, A2 B2 and A3 B3 or Type 2. A B A B, A2 B2 and A3 B3 Proposition 2.5[7] For any neutrosophi risp set A the following are hold φn A, φn φn A XN, XN XN 36

3 New Trends in Neutrosophi Theory and Appliations Definition 2.6[7] Let Xbe a non-empty set, and the two NCSsA and Bgiven in the form A = (A, A2, A3), B = (B, B2, B3), then : ) A B may be defined as two types: i)type. A B = A B, A2 B2, A3 B3 ii) Type 2. A B = A B, A2 B2, A3 B3 2) A B may be defined as two types: i) Type. A B = A B, A2 B2, A3 B3 ii) Type 2. A B = A B, A2 B2, A3 B3 Definition 2.7[7] A neutrosophi risp topology (NCT ) on a non-empty set X is a family Γof neutrosophi risp subsets ofx satisfying the following axioms: i) N, XN Γ. ii) A A 2 Γ, A, A 2 Γ. iii) A j Γ, {A j : j J} Γ. In this ase, the pair (X, Γ) is alled a neutrosophi risp topologial spae (NCTS) in X. The elements of Γare alled neutrosophi risp open sets (NCOSs) in X. A neutrosophi risp set F is losed if and only if its omplement F is an open neutrosophi risp set. Definition 2.8[7] Let (X, Γ) be NCTS and A = A, A2, A3 be a NCS in X. Then the neutrosophi risp losure of A (NCl(A)) and neutrosophi interior risp (NCint(A) ) of A are defined by NCl(A)= {K:K is an NCCS in X and A K} NCint (A) = {G:G is an NCOS in X and G A), Where NCS is a neutrosophi risp set and NCOS is a neutrosophi risp open set. It an be also shown that NCl(A) is a NCCS (neutrosophi risp losed set) and NCint(A) is a NCOS (neutrosophi risp open set) in X. 3. Neutrosophi Crisp Nearly Open Sets Definition 3. Let (X, Γ) be a NCTS and A = A, A2, A3 be a NCS in X, then A is alled: Neutrosophi risp α-open set iffa NCint(NCl(NCint(A)). [24] i) Neutrosophi risp pre-open set iffa NCint(NCl(A)). ii) Neutrosophi risp semi-open set iffa NCl(NCint (A)). iii) Neutrosophi risp β- open set iffa (NCl (NCint(NCl (A)). We shall denote the lass of all neutrosophi risp α- open sets asncγ α, and the lass of all neutrosophi risp pre-open sets as NCΓ p, and the lass of all neutrosophi risp semiopen sets as NCΓ S, and the lass of all neutrosophi risp β- open sets as NCΓ β. Definition 3.2 Let (X, Γ) be a NCTS and B = B, B2, B3 be a NCS in X, then B is alled: i) Neutrosophi risp α-losed set iff (NCl (NCint(NCl (B)) B. ii) Neutrosophi risp pre- losed set iff NCl(NCint (B)) B. iii) Neutrosophi risp semi- losed set iff NCint(NCl(B)) B. iv) Neutrosophi risp β- losed set iff NCint(NCl(NCint(B)) B. One an easily show that, the omplement of a neutrosophi risp (α, pre, semi, β)- open set is a neutrosophi risp (α, pre, semi, β)- losed set,respetively. 37

4 Florentin Smarandahe, Surapati Pramanik (Editors) Remark 3.3 For the lass onsisting of exatly all a NCα- struture and NCβ- struture, evidently, NCΓ NCΓ α NCΓ β. We notie that every non-empty NCβ- open has NCα-open non-empty interior. If all neutrosophi risp sets the family {Bi}i I, are NC β- open sets, then Proposition 3.4 Consider, { Bi}i I, is a family of NCβ- open sets, then { Bi}i I NCl(NCint(Bi)) NCl(NCint(Bi)), that is A NCβ- struture is a neutrosophi losed with respet to arbitrary neutrosophi risp unions. We shall now haraterize NCΓ α in terms ofncγ β. Definition 3.5 Let (X, Γ) be a NCTS and A = A, A2, A3 be a NCS in X, then: NClα(A) = { G:G A and G is NCα-losed} NCintα (A) = {G:G A and G is NCα- open} NCl pre (A) = { G:G A and G is NCpre-losed} NCint pre (A) = {G:G A and G is NCpre- open} Definition 3.6 NCl semi (A) = { G:G A and G is NCsemi-losed} NCint semi (A) = {G:G A and G is NCsemi- open} NClβ(A) = { G:G A and G is NCβ-losed} NCintβ(A) = {G:G A and G is NCβ- open} Theorem 3.7 Let (X, Γ) be a NCTS. NCΓ α Consists of exatly those NCSA for whih A B NCΓ β for B NCΓ β. Let A NCΓ α, B NCΓ β, P A B and U be a neutrosophi risp neighborhood (for short NCnbd) of p. Clearly U NCint(NCl(NCint(A)), too is a neutrosophi risp open neighborhood of P, so V=(U NCint(NCl(NCint(A)))) NCint(B) is non-empty. Sine V NCl(NCint(A)) this implies (U NCint(A) NCint(B) =V NCint(A) = N. It follows that Conversely, A B NCl(NCint(A) NCint(B))= NCl(NCint(A B)) i.e. A B NCΓ β. Let A B NCΓ β for all B NCΓ β. then in partiular A NCΓ β. Assume that P A (NCint(NCl(A) (NCint(A))). Then P NCl(B), where (NCl(NCint(A))) Clearly {P} B NCΓ β and onsequently A {{P} B} NCΓ β. But A {{P} B}={P}. Hene {P} is a neutrosophi risp open. P (NCl(NCint(A))) implies P )NCint(NCl(NCint(A)()), ontrary to assumption. Thus P A implies P (NCl(NCint(A)) and A NCΓ α. Thus we have found that NCΓ α is omplete determined by NCΓ β i.e. all neutrosophi risp topologies with the same NCβ- struture also determined the same NCα-struture, expliitly given Theorem 3.. We shall prove that onversely all neutrosophi risp topologies with the same NCαstruture, so that NCΓ β, is ompletely determined by NCΓ α Theorem 3.8 Every NCα-struture is a NCΓ. 38

5 New Trends in Neutrosophi Theory and Appliations NCΓ β Contains the neutrosophi risp empty set and is losed with respet to arbitrary unions. A standard result gives the lass of those neutrosophi risp sets A for whih A B NCΓ β for all B NCΓ β onstitutes a neutrosophi risp topology, hene the theorem. We may now haraterize NCΓ β, in terms of NCΓ α in the following way. Proposition 3.9 Let (X, Γ) be a NCTS. Then NCΓ β = NCΓ αβ and hene NCα -equivalent topologies determine the same NCβ -struture. Let NCα l andα int denote neutrosophi losure and Neutrosophi risp interior with respet to NCΓ α. If P B NCΓ β and P B NCΓ α, then (NCint(NCl(NCint(A))) NCint(B)) N. Sine (NCint(NCl(NCint(A))) is a risp neutrosophi neighbor-hood of point p, so ertainly NCint(B) meets NCl(NCint(A)) and therefore (big neutrosophi open) meets NCint(A), proving A NCint(B) N. This means B NCαl(NCint(B)).i.e. B NCΓ αβ on the other hand let A NCΓ αβ, P A. and P V NCΓ. As V NCΓ α, and P NCl(NCint(A)), we have V NCint(A) N and there exist a neutrosophi trip set W Γ suh that W V NCαint(A) A. In other words V (NCint(A)) N and P NCl(NCint(A)). Thus we have verified NCΓ αβ NCΓ α, and the proof is omplete ombining Theorem 3. and Proposition 3.. and we get NCΓ αα = NCΓ α. Corollary 3.0 A neutrosophi risp topology NCΓis a NCα - topology iff NCΓ = NCΓ α. Evidently NCΓ β is a neutrosophi risp topology iffncγ α = NCΓ β. In this ase NCΓ ββ = NCΓ αβ = NCΓ β. Corollary 3. NCβ-Struture B is a neutrosophi risp topology, then B= Bα= Bβ. We proeed to give some results on the neutrosophi struture of neutrosophi risp NCα topology Proposition 3.2 The NCα-open with respet to a given neutrosophi risp topology are exatly those sets whih may be written as a differene between a neutrosophi risp open set and a neutrosophi risp nowhere dense set. If A NCΓ α we have A= NCint(NCl(NCint(A))) (NCint(NCl(NCint(A)) A ), where (NCint(NCl(NCint(A)) A ) learly is neutrosophi risp nowhere dense set, we easily see that B NCl(NCint(A)) and onsequently A B NCint(NCl(NCint(A)) so the proof is omplete. Corollary 3.3 A neutrosophi risp topology is a NCα- topology iff all neutrosophi risp nowhere dense sets are neutrosophi risp losed. For a neutrosophi risp NCα-topology may be haraterized as neutrosophi risp topology where the differene between neutrosophi risp open and neutrosophi risp nowhere dense set is again a neutrosophi risp open, and this evidently is equivalent to the ondition stated. Proposition 3.4 Neutrosophi risp topologies whih are NCα- equivalent, determine the same lass of neutrosophi risp nowhere dense sets. 39

6 Florentin Smarandahe, Surapati Pramanik (Editors) Proposition 3.6 If a NCα -Struture B, is a neutrosophi risp topology, then all neutrosophi risp topologies Γ for whih Γ β = B are neutrosophi risp extremely disonneted. In partiular: Either all or none of the neutrosophi risp topologies of a NCα lass are extremely disonneted. Let Γ β = B, and suppose there is A Γ suh that NCl(A) Γ. Let P NCl(A) NCint(NCl(A)) with B = {P} NCint(NCl(A)), Μ = NCint(NCl(A)) We have {P} Μ= (NCint(NCl(A)) = NCl(NCint(Μ)), {P} NCl(A) = NCl(NCint(NCl(A)) NCl(NCint(B)). Hene both B and Μ are in Γ β. The intersetion B Μ= {P} is not neutrosophi risp open, sine P NCl(A) Μ hene not NCβ- open. So, Γ β =B is not a neutrosophi risp topology. Now suppose B is not a topology, and Γ β =B There is a B Γ β suh that B Γ α. Assume that NCl(NCint(B)) Γ. Then B NCl(NCint(B))=NCint(NCl(NCint(B)) i.e. B Γ α, ontrary to assumption. Thus we have produed an open set whose losure is not open, whih ompletes the proof. Corollary 3.7 A neutrosophi risp topology Γ is a neutrosophi risp extremally disonneted if and only if Γ β is a neutrosophi risp topology. Remark 3.8 The following diagram represents the relation between neutrosophi risp nearly open sets: NCpre- NC- NCα- NCβ- NCsemi- 4. Neutrosophi Crisp Continuity We, introdue and study of neutrosophi risp ontinuous funtion and we obtain some haraterizations of neutrosophi ontinuity. Here ome the basi definitions first: Definition 4. Let (X, Γ) be a NCTS and A = A,A2,A3 be a NCS in X, and f: X X then: ) If fncα-ontinuous inverse image of NCα open set is NCα- open set 2) If f NCpre-ontinuous inverse image of NCpre-open set is NCpre- open set 3) If f NCsemi-ontinuous inverse image of NCsemi-open set is NCsemi- open set 4) If f NCβ-ontinuous inverse image of NCβ-open set is NCβ- open set Definition 4.2 The following was given in [24] (a) If A A, A2, A3 is a NCS in X, then the neutrosophi risp image of A under f, 320

7 New Trends in Neutrosophi Theory and Appliations denoted by f (A), is the a NCS in Y defined by f ( A) f ( A ), f ( A2 ), f ( A3 ). (b) If f is a bijetive map then f - : Y X is a map defined suh that: for any NCS B B, B2, B3 in Y, the neutrosophi risp preimage of B, denoted by ( B ), ( 2 3 f is a NCSin X defined by f B) f ( B ), f ( B ), f ( B ). Definition 4.3 Let (X, Γ), and (Y, Γ2) be two NCTSs, and let f: X Y be a funtion. Then f is said to be ontinuous if f the preimage of eah NCS in Γ2 is a NCS in Γ. Definition 4.4 Let (X, Γ), and (Y, Γ2) be two NCTSs and let f: X Y be a funtion. Then f is said to be open iff the image of eah NCS in Γ, is a NCS in Γ2. Proposition4.5 Let X, o and Y, o be two NCTSs. If f : X Y is ontinuous in the usual sense, then in this ase, f is ontinuous in the sense of Definition 4.3 too. Here we onsider the NCTSson X and Y, respetively, as follows: G,, G : Go and H,, H : H 2, o In this ase we have, for eah H,, H, 2 H, o f H,, H f ( H), f ( ), f ( H ) f H,,( f ( H)). Now we obtain some haraterizations of neutrosophi ontinuity. Proposition 4.6 Let f: (X, Γ) (Y, Γ2). Then f is neutrosophi ontinuous iff the preimage of eah neutrosophi risp losed set (NCCS) in Γ2 is a NCCS in Γ. Proposition 4.7 The following are equivalent to eah other: (a) f: (X, Γ) (Y, Γ2) is neutrosophi ontinuous. (b) f - (NCint(B) NCint(f - (B))) for eah NCSBin Y. () (NCl f - (B)) f - (NCl(B)). for eah NCSB in Y. Corollary 4.8 Consider ( X, ) and Y, 2 to be two NCTSs, and let f : X Y be a funtion. if f ( H ) : H 2. Then will be the oarsest NCT on X whih makes the funtion f : X Y ontinuous. One may all it the initial neutrosophi risp topology with respet to f. 5. Neutrosophi Crisp Compat Spae First we present the basi onepts: Definition5. Let X, be an NCTS. (a) If a family Gi : i J 2 3 G G, G : i J X, i, i2 i3 N of NCOSs in X satisfies the ondition then it is alled an neutrosophi open over of X. (b) A finite subfamily of an open over G, G, G : i J i i i on X, whih is also a

8 Florentin Smarandahe, Surapati Pramanik (Editors) neutrosophi open over of X, is alled a neutrosophi risp finite open subover. Definition5.2 A neutrosophi risp set A A, A2, A3 in a NCTS X, is alled neutrosophi risp ompat iff every neutrosophi risp open over of A has a finite neutrosophi risp open subover. Definition5.3 A family Ki, Ki, Ki : i J of neutrosophi risp ompat sets in X satisfies the finite 2 3 intersetion property (FIP ) iff every finite subfamily Ki, Ki, Ki : i,2,..., n of the family 2 3 K K, K : i,2,..., n. satisfies the ondition i i i N 322, 2 3 Definition5.4 A NCTS X,is alled neutrosophi risp ompat iff eah neutrosophi risp open over of X has a finite open subover. Corollary5.5 A NCTS, of X is a neutrosophi risp ompat iff every family G G, G : i J i, i2 i3 neutrosophi risp ompat sets in X having the finite intersetion properties has nonempty intersetion. Corollary5.6 Let X,, Y, 2 be NCTSs and f : X Y be a ontinuous surjetion. If X, is a neutrosophi risp ompat, then so is Y, 2. Definition5.7 If a family Gi : i J of neutrosophi risp ompat sets in X satisfies the 2 3 ondition A G, G, G : i J i i i, then it is alled a neutrosophi risp open over of A. 2 3 Let s onsider a finite subfamily of a neutrosophi risp open subover of Gi : i J. 2 3 Corollary5.8 Let X,, Y, 2 be NCTSs and f : X Y be a ontinuous surjetion. If A is a neutrosophi risp ompat in X, then so is (A) Y., f in 6. Conlusion In this paper, we presented a generalization of the neutrosophi topologial spae. The basi definitions of the neutrosophi risp topologial spae and the neutrosophi risp ompat spae with some of their haraterizations were dedued. Furthermore, we onstruted a neutrosophi risp ontinuous funtion, with a study of a number its properties. Referenes. K. Atanassov, intuitionisti fuzzy sets, Fuzzy Sets and Systems 20, 87-96,(986). 2. C.L.Chang, Fuzzy Topologial Spaes,. Math. Anal-Appl. 245,82-90(968). 3. I.M. Hanafy, A.A. Salama and K.M. Mahfouz, Neutrosophi Crisp Events and Its Probability, International Journal of Mathematis and Computer Appliations Researh(IJMCAR) Vol.(3), Issue, pp.7-78,mar (203). 4. I. M. Hanafy, A.A. Salama and K. Mahfouz, Correlation of Neutrosophi Data, International Refereed Journal of Engineering and Siene (IRJES), Vol.(), Issue 2. PP (202)., 2

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