Smooth Neutrosophic Topological Spaces

Transcription

1 65 Unversty of New Mexco Smooth Neutrosophc opologcal Spaces M. K. EL Gayyar Physcs and Mathematcal Engneerng Dept., aculty of Engneerng, Port-Sad Unversty, Egypt.- Abstract. As a new branch of phlosophy, the neutrosophy was presented by Smarandache n 980. t was presented as the study of orgn, nature, and scope of neutraltes; as well as ther nteractons wth dfferent deatonal spectra. he am n ths paper s to ntroduce the concepts of smooth neutrosophc topologcal space, smooth neutrosophc cotopologcal space, smooth neutrosophc closure, and smooth neutrosophc nteror. urthermore, some propertes of these concepts wll be nvestgated. Keywords: uzzy Sets, Neutrosophc Sets, Smooth Neutrosophc opology, Smooth Neutrosophc Cotopology, Smooth Neutrosophc Closure, Smooth Neutrosophc nteror. ntroducton n 986, Badard [] ntroduced the concept of a smooth topologcal space as a generalzaton of the classcal topologcal spaces as well as the Chang fuzzy topology []. he smooth topologcal space was redscovered by Ramadan [3], and El-Gayyar et al. [4]. n [5], the authors ntroduced the notons of smooth nteror and smooth closure. n 983 the ntutonstc fuzzy set was ntroduced by Atanassov [[6], [7], [8]], as a generalzaton of fuzzy sets n Zadeh s sense [9], where besdes the degree of membershp of each element there was consdered a degree of non-membershp. Smarandache [[0], [], []], defned the noton of neutrosophc set, whch s a generalzaton of Zadeh s fuzzy sets and Atanassov s ntutonstc fuzzy set. he words neutrosophy and neutrosophc were nvented by. Smarandache n hs 998 book. Etymologcally, neutro-sophy (noun) [rench neutre < Latn neuter, neutral, and Greek sopha, skll/wsdom] means knowledge of neutral thought. Whle neutrosophc (adjectve), means havng the nature of, or havng the characterstc of Neutrosophy. Neutrosophc sets have been nvestgated by Salama et al. [[3], [4], [5]]. he purpose of ths paper s to ntroduce the concepts of smooth neutrosophc topologcal space, smooth neutrosophc cotopologcal space, smooth neutrosophc closure, and smooth neutrosophc nteror. We also nvestgate some of ther propertes. PRELMNARES n ths secton we use to denote a nonempty set, to denote the closed unt nterval [ 0, ], o to denote the nterval ( 0, ], to denote the nterval [ 0, ), and to be the set of all fuzzy subsets defned on. By 0 and we denote the characterstc functons of and, respectvely. he famly of all neutrosophc sets n wll be denoted by ().. Defnton [], [], [5] A neutrosophc set A (NS for short) on a nonempty set s defned as: A x,a,a,a, x where,,: [0, ], and 0 A A A 3 representng the degree of membershp (namely A ), the degree of ndetermnacy (namely A ), and the degree of non-membershp (namely A ); for each element x to the set A..Defnton [3], [4], [5] he Null (empty) neutrosophc set 0 N and the absolute (unverse) neutrosophc set N are defned as follows: ype : 0N x,0,0,, x ype : 0N x,0,,, x ype : N x,,,0, x ype : N x,,0,0, x M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

2 66.3Defnton [3], [4], [5] A neutrosophc set A s a subset of a neutrosophc set B, ( A B), may be defned as: ype ype : A B A B, A B,A B, : A B A B, A B,A B,.4Defnton [3], [4], [5] x x he Complement of a neutrosophc set A, denoted by coa, s defned as: ype : coa ype : coa x,a, A,A x, A, A, A.5Defnton [3], [4], [5] Let A,B() then: ype ype ype ype : A J : A J : A J : A J.7Defnton [3], [4], [5] x,sup A,sup,nf A A j j j x,sup A,nf,nf A A j j j x,nf A,nf,sup A A j j j x,nf A,sup,sup A A j j j he dfference between two neutrosophc sets A and B defned as A \ B A cob..8defnton [3], [4] Every ntutonstc set A on s NS havng the form A x,a, (A A ),A, and every fuzzy set A on s NS havng the form A x,,0,, x. A A ype : ype : ype : ype : A B x, max( A,B ), max( A,B),mn(A,B ) A B x, max( A,B ), mn( A,B),mn(A,B ) A B x, mn(a,b ), mn( A,B),max( A,B ) A B x, mn(a,b ), max( A,B),max( A,B ).9Defnton [5] Let Y be a subset of and A on Y s denoted by A / Y A extenson of B on, denoted by B B 0.5.0Defnton [],[3]. or each ; the restrcton of Y B, the B, s defned by: f x A f Y A smooth topologcal space (SS, for short) s an ordered par (, ) where s a nonempty set and : s a mappng satsfyng the followng propertes: [ ]A x,a,a, A A x, A,A,A.6Defnton [3], [4], [5] Let { A }, J be an arbtrary famly of neutrosophc sets, then: (O) (O) (O3) (0) () A,A, (A A ) (A ) (A ) A, J, ( A ) (A ) J J.Defnton [],[3] A smooth cotopology s defned as a mappng : whch satsfes: M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

3 67 (C) (C) (C3) (0) () B,B, (B B ) (B ) (B ) A, J, ( B ) (B ) J J 3.Smooth Neutrosophc opologcal spaces we wll defne two types of smooth neutrosophc topologcal spaces, a smooth neutrosophc topologcal space (SNS, for short) take the form (,,, ) and the mappngs,, : represent the degree of openness, the degree of ndetermnacy, and the degree of non-openness respectvely. 3. Smooth Neutrosophc opologcal spaces of type n ths part we wll consder the defntons of type. 3..Defnton A smooth neutrosophc topology (,, ) of type satsfyng the followng axoms: (SNO ) (0) (0) () (), and (0) () 0 (SNO ) A,A, (A A), (A A),and (A A) (SNO 3) A, J, ( A), J J ( A),and J J ( A) J J 3..Defnton Let,, : followng axoms: be mappngs satsfyng the (SNC ) (0) (0) () (), and (0) () 0 (SNC ) B,B, (B B) (B) (B), (B B) (B) (B),and (B B) (B) (B) (SNC 3) B, J, ( B ) (B ), J J ( B ) (B ),and J J ( B ) (B ) J J he trple (,, ) s a smooth neutrosophc cotopology of type,,, represent the degree of closedness, the degree of ndetermnacy, and the degree of non-closedness respectvely. 3..3Example Let {a,b}.defne the mappngs,, : as: f A 0 f A mn(a(a),a(b)) f f A 0 f A 0.5 f 0 f A 0 0 f A max(a(a),a(b)) f A snether 0 nor A snether 0 nor A snether 0 nor hen (,,, ) s a smooth neutrosophc topologcal space on. 3..4Proposton Let (,, ) and (,, ) be a smooth neutrosophc topology and a smooth neutrosophc cotopology, respectvely, and let A, M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

4 68 (coa), (coa), (coa), (coa), (coa), and (coa), then () (,, ) and (,, ) are a smooth neutrosophc topology and a smooth neutrosophc cotopology, respectvely. (),,, Proof,,, () (a) (0) () (0) (), and (b) (0) () 0 A,A, (A A ) (co(a A) (coa coa) (coa ) (coa) (A ),smlarly, A,A, ( A A),and ( A A) (c) A, J, ( A) (co A) J J ( coa ) (coa ) J J J,smlarly, A,J, ( A),and J J ( A ) (A ).Hence, (,, ) J J s a smooth neutrosophc topology.smlarly, we can prove that (,, ) s a smooth neutrosophc cotopology. () the proof s straghtforward. 3..5Proposton Let{(,, )} J be a famly of smooth neutrosophc topologes on.hen ther ntersecton (,, ) J s a a smooth neutrosophc topology. Proof he proof s a straghtforward result of both defnton(.6) and dfnton (3..). 3..6Defnton Let (,, ) be a smooth neutrosophc topology of type, and A. hen the smooth neutrosophc closure of A, denoted by A s defned by: A,( ) (,,0) {H : H,A H, (H) A (H) (H) }, ( ) (,,0) 3..7Proposton Let (,, ) be a smooth neutrosophc topology on, and A,B. hen () 0 0, () A A (3), and ( A) }, A (4) B A, (B), (B) and ( A) (B) B A, A,B M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

5 69 (5) A A (6) A B A B ths famly contans A, hence, (c) f B B, and A A B A A Proof () Obvous () Drectly from defnton (3..6) (3) (a) f A A, the proof s straghtforward. (b) f A A, we have from the defnton (3..) and the defnton (3..6): rom defnton (3..6) every element n the famly A wll be an element n the famly B, hence B A. (5) rom (), (3) and the defnton (3..6) we have A A. (6) (a) f A A, and B B, then A B A B A B A B (H) (H) (H) (H) ( {H : H }) { } (H) (H),A H, (H) : H,A H, can prove that n a smlar way. (H) (H) (H) (H) ( {H : H }) { } (H) (H),A H, (H) : H,A H, we (b) f (c) f (d) f (e) f A A, B B, and A B A B, from (4) B A B, hence A B A B A A, B B, and A B A B, then A A B, hence A B A B A A, B B, and A B A B, smlar to(6b) A A, B B, and A B A B, smlar to(6c) (f) f A A, B B,and A B A B, t follows from(4)that A A B, hence A B A B. (4) (a) f B B, then A A and B A. (b) f B B, and A A B {H : H (H) (B),,B H, (H) (H) (B)}, (B), (g) f A A, B B,and A B A B M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

6 70 A B {H : H,A B H, (H) (A B), (H) (A B), (H) (A B)} {H : H,A B H, (H) (B), (H) (B), (H) (B)} {H : H,A H, B H, (H) or (H) (B), (H) or (H) (B), (H) or (H) (B)} [ {H : H,A H, (H) (H) (H) } {H : H,B H, (H) (B), (H) (B), (H) (B)} ] [ {H : H,A H, (H) (H) (H) } ] [ {H : H,B H, (H) (B), (H) (B), (H) (B)} ] A B 3..8Defnton Let (,, ) be a smooth neutrosophc topology of type, and A. hen the smooth neutrosophc nteror of A, denoted by o A s defned by: A,( ) (,,0) o {H : H,H A, (H) A (H) (H) }, ( ) (,,0) 3..9Proposton Let (,, ) be a smooth neutrosophc topology on, and A,B. hen o o () 0 0, () A o A o o (3) (A ) (A ), and o ( A ) }, A (4) B A, (B) (B) o o and ( B) B A, A,B o o o (5) ( A ) A o o o (6) ( A B) A B Proof Smlar to the procedure used to prove Proposton (3..7) 3.. Smooth Neutrosophc opologcal spaces of type n ths part we wll consder the defntons of type. n a smlar way as n type, we can state the followng defntons and propostons. he proofs of the propostons of type, wll be smlar to the proofs of the propostons n type. M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

7 7 3..Defnton A smooth neutrosophc topology (,, ) of type satsfyng the followng axoms: (SNO ) (0) (), and (0) () (0) () 0 (SNO ) A,A, (A A), (A A),and (A A) (SNO 3) A, J, ( A), J J ( A),and J J ( A) J J 3..Defnton Let,, : be mappngs satsfyng the followng axoms: (SNC ) (0) (), and (0) () (0) () 0 (SNC ) B,B, (B B) (B) (B), (B B) (B) (B),and (B B) (B) (B) (SNC 3) B, J, ( B ) (B ), J J ( B ) (B ),and J J ( B ) (B ) J J he trple (,, ) s a smooth neutrosophc cotopology of type,,, represent the degree of closedness, the degree of ndetermnacy, and the degree of non-closedness respectvely. 3..3Example Let {a,b}. Defne the mappngs,, : as: f A 0 f A mn(a(a),a(b)) f 0 f A 0 0 f A 0.5 f 0 f A 0 0 f A max(a(a),a(b)) f A snether 0 nor A snether 0 nor A snether 0 nor hen (,,, ) s a smooth neutrosophc topologcal space on. Note that: the Propostons (3..4) and (3..5) are satsfed for type. 3..4Defnton Let (,, ) be a smooth neutrosophc topology of type, and A. hen the smooth neutrosophc closure of A, denoted by A s defned by: A,( ) (,,0) {H : H,A H, (H) A (H) (H) }, ( ) (,,0) Also, the smooth neutrosophc nteror of A, denoted by o A s defned by: A,( ) (,,0) o {H : H,H A, (H) A (H) (H) }, ( ) (,,0) Note hat: the Propostons (3..7) and (3..9) are satsfed for type. 4. Concluson and uture Work M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces

8 7 n ths paper, the concepts of smooth neutrosophc topologcal structures were ntroduced. n two dfferent types we ve presented the concepts of smooth neutrosophc topologcal space, smooth neutrosophc cotopologcal space, smooth neutrosophc closure, and smooth neutrosophc nteror. Due to unawareness of the behavour of the degree of ndetermnacy, we ve chosen for to act lke n the frst type, whle n the second type we preferred that behaves lke. herefore, the defntons gven above can also be modfed n several ways dependng on the behavour of. Moreover, as a consequence of our choces of the performance of and, one can see that: n type, booth defned n (3..) wth ther condtons are smooth topologes; whle n type, only defned n (3..) wth ts condtons s a smooth topology. Acknowledgement would lke to express my worm thanks to Prof. Dr. E. E. Kerre for hs valuable dscutons and to Prof. Dr. A. A. Ramadan, who ntroduced me to the world of smooth structures. We thank Prof. Dr. lorentne Smarandache [Department of Mathematcs, Unversty of New Mexco, USA], Prof. Dr. Ahmed Salama [Department of Mathematcs and Computer Scence, aculty of Scences, Port Sad Unversty, Egypt] for helpng us to understand neutrosophc approach. References [] R. Badard, Smooth axomatcs, st SA Congress, Palma de Mallorca, 986. [] C. Chang, uzzy topologcal spaces, J. Math. Anal. Appl. 4 (968) [3] A. Ramadan, Smooth topologcal spaces, uzzy Sets and Systems 48 (99) uzzy Logc and echnology EUSLA 003, Zttau (003) 9. [7] K.. Atanassov, ntutonstc fuzzy sets, uzzy Sets and Systems 0 (986) [8] C. Cornels, K.. Atanassov, E. E. Kerre, ntutonstc fuzzy sets and nterval-valued fuzzy sets: A crtcal comparson, Proc. EUSLA03 (003) [9] L. A. Zadeh, uzzy sets, nform. and Control 8 (965) [0]. Smarandache, Neutrosophy and neutrosophc logc, n: rst nternatonal Conference on Neutrosophy, Neutrosophc Logc, Set, Probablty, and Statstcs, Unversty of New Mexco, Gallup, NM 8730,USA. []. Smarandache, A Unfyng eld n Logcs: Neutrosophc Logc. Neutrosophy, Neutrosophc crsp Set, Neutrosophc Probablty, Amercan Research Press, 999. []. Smarandache, Neutrosophc set, a generalzaton of the ntutuonstcs fuzzy sets, nter. J. Pure Appl. Math., 4 (005) [3] A. A. Salama, S. Alblow, generalzed neutrosophc set and generalzed neutrosophc topologcal spaces, Journal Computer Sc. Engneerng (0) 9 3. [4] A. A. Salama,. Smarandache, S. Alblow, New neutrosophc crsp topologcal concepts, Neutrosophc Sets and Systems (04) [5] A.A. Salama and lorentn Smarandache, Neutrosophc crsp set theory, Educatonal Publsher, Columbus, (05).USA Receved: ebruary 3, 06. Accepted: June 0, 06. [4] M. El-Gayyar, E. Kerre, A. Ramadan, On smooth topologcal spaces : separaton axoms, uzzy Sets Syst. 9 (00) [5] M. El-Gayyar, E. Kerre, A. Ramadan, Almost compactness and near compactness n smooth topologcal spaces, uzzy Sets and Systems 6 (994) [6] K.. Atanassov, ntutonstc fuzzy sets: past, present and future, Proc. of the hrd Conf. of the European Socety for M. K. EL Gayyar, Smooth Neutrosophc opologcal Spaces