Second Order Fuzzy S-Hausdorff Spaces

Transcription

1 Inten J Fuzzy Mathematical Achive Vol 1, 013, ISSN: (P), (online) Publihed on 9 Febuay 013 wwweeachmathciog Intenational Jounal o Second Ode Fuzzy S-Haudo Space AKalaichelvi Depatment o Mathematic Avinahilingam Intitute o Home Science and Highe Education o Women Coimbatoe kalaichelviadu@yahoocom Received 6 Decembe 01; accepted 7Januay 013 Abtact In thi pape, the deinition o S-Haudo pace intoduced by Sivatava, R, Lal, SN and Sivatava, AK [6] i extended to econd ode uzzy topological pace in two dieent way It i hown that thee two concept ae heeditay and poductive The elation between it ode and econd ode concept ae tudied The behavio o thee concept with egad to i( δ ), i ( δ ), i ( δ ), ω ( ), ( ω ( )) and ( ) ae alo analyed ω AMS Mathematic Subject Claiication (010): 94D05 Keywod: econd ode uzzy et, S-Haudo Space 1 Intoduction A uzzy et on a et X i a map deined on X with value in Ι, whee Ι i the cloed unit inteval [0, 1] Equivalently uzzy et which ae named a it ode uzzy et in thi tudy deal with cipy deined membehip unction o degee o membehip It i doubtul whethe, o intance, human being have o can have a cip image o membehip unction in thei mind Zadeh [8] theeoe uggeted the notion o a uzzy et whoe membehip unction itel i a uzzy et Thi lead to the ollowing deinition o a econd ode uzzy et o a uzzy et o type A econd ode uzzy et on a nonempty et X i a map om X to Ι Ι Fit ode uzzy et ae denoted by, g, h, and econd ode uzzy et ae denoted by, g, h, In thi pape the tem uzzy et and it ode uzzy et ae ued ynonymouly Wheneve a uzzy et i conideed without mentioning the ode, it alway ee to a it ode uzzy et Simila teminology applie to all concept elated to it ode uzzy et 41

2 AKalaichelvi Fundamental deinition and popetie o econd ode uzzy et and econd ode uzzy topological pace ae intoduced in [4] Six impotant and inteeting connection c/ 1, c/, c/ 3, c/ 4, c/ 5 and c/ 6 between it ode and econd ode uzzy topological pace ae alo dicued in [4] The concept o uzzy topological pace wa intoduced and developed by Chang[1]and o othe application one may ee[& 3]Connection between cip topological pace and econd ode uzzy topological pace ae given in [5] With evey cip topology on a nonempty ex X, thee econd ode uzzy topologie ω ( ), ( ω ( )) and ω ( ) on X ae aociated Alo with evey econd ode uzzy topology δ on a nonempty et X, thee cip topologie i( δ ), i ( δ ) and i ( δ ) on X ae aociated [5] In thi pape the deinition o S-Haudo pace intoduced by Sivatava, R, Lal, SN and Sivatava, AK [6] i extended to econd ode uzzy topological pace in two dieent way It i hown that thee two concept ae heeditay and poductive The elation between it ode and econd ode concept ae tudied The behavio o thee concept with egad to i( δ ), i ( δ ), i ( δ ), ω ( ), ( ω ( )) and ω ( ) ae alo analyed Fundamental Deinition And Notation Deinition 1 [4] A econd ode Chang uzzy topology δ on a nonempty et X i a collection o econd ode uzzy et on X atiying the ollowing condition : (i) (ii) (iii) 0, 1 δ whee, o any x X 0 (x) = the zeo unction 0 and Ι λ δ o each λ Λ implie ( λ ) δ Λ m i δ o i = 1,,, m implie ( i ) δ i = 1 1(x) = the contant unction 1 on Ι λ The pai (X, δ ) i called a econd ode Chang uzzy topological pace A econd ode Lowen uzzy topology δ on X i deined by eplacing axiom (i) in the above deinition by axiom (i) (i) All contant unction α δ whee, o any x X, α (x) = the contant unction α on Ι The pai (X, δ ) i called a econd ode Lowen uzzy topological pace The element o δ ae called econd ode uzzy open et Deinition [4] Evey it ode uzzy topology δ = { λ / λ Λ} on a non-empty et X deine a econd ode uzzy topology δ = { λ / λ δ} on X whee λ (x) (α) 4

3 Second Ode Fuzzy S-Haudo Space = λ (x), o evey x X and o evey α Ι The coepondence δ δ i denoted a c/ 1 Let δ = { λ / λ Λ} be a econd ode uzzy topology (Lowen) on X Fix α Ι The collection δ α = ditinct element o the collection { ( λ ) / λ δ } deine a it ode uzzy topology on X whee (λ ) α (x) = λ (x)(α), o evey x X c/ 3 denote the coepondence δ δ α The collection S = {( λ ) / λ δ } i a ubbae o a it ode uzzy topology δ on X whee ( λ ) (x) = λ (x)(α), o evey x X c/ 4 denote the α Ι coepondence δ δ Given a econd ode uzzy topology on a nonempty et X, the aociation c/ 6, give a way o getting anothe econd ode uzzy topology on the ame et X That i, given a econd ode uzzy topology δ = { λ /λ Λ} on a nonempty et X, the collection δ c = {( λ )c / λ δ } i alo a econd ode uzzy topology on X Deinition 3 [5] Let (X, δ ) be a econd ode uzzy topological pace Deine (1) i ( δ ) to be the topology geneated by the collection {( {(A ) / δ} 1 whee o (0, 1), (A ) = {x X / (x) (, 1] = Ι} () i*( δ ) to be the topology geneated by the collection {A / δ } whee (A ) = {x X / (x) 1 (, 1] = Ι o ome (0, 1)} (3) i( δ ) to be the topology having the collection {(A ) / δ, (0,1) } a a ub bai Deinition 4 [5] Let (X, ) be a topological pace Deine (1) (ω ()) to be the econd ode uzzy topology geneated by K whee Ι X o (0, 1), K = { ( Ι ) /(A ) } () ω () to be the econd ode uzzy topology geneated by K whee K = { ( Ι Ι ) X /(A ), o evey (0, 1)} (3) ω ( ) to be the econd ode uzzy topology geneated by K whee { ( Ι Ι ) X / A } α K = 43

4 AKalaichelvi Deinition 5 [6] A uzzy topological pace (X, δ) i aid to be uzzy S-Haudo i o any pai o ditinct uzzy point x t, y in X, thee exit, g δ uch that x t, y g and Λ g = 0 3 Second Ode Fuzzy S-Haudo Space Deinition 31 A econd ode uzzy topological pace (X, δ ) i aid to be econd ode uzzy S-Haudo o type 1, denoted by (S H) 1, i o any pai o ditinct econd ode uzzy point x, y in X, thee exit, g δ uch that x, y g and Λ 1 g = 0, whee Λ 1 g = 0 mean given x X, eithe (x) = 0 o g(x) = 0 Deinition 3 A econd ode uzzy topological pace (X, δ ) i aid to be econd ode uzzy S-Haudo o type, denoted by (S H), i deined by eplacing the condition Λ 1 g = 0 in the above deinition by Λ g = 0, whee Λ g = 0 mean given x X and α Ι eithe (x) (α) = 0 o g(x) (α) = 0 Note: (i) (X, δ ) i (S H) 1 (X, δ ) i (S H) But the convee need not be tue (ii) Subpace o a (S H) 1 pace i (S H) 1 Deinition 33 Let ( X, δ 1), ( Y, δ ) be two econd ode uzyz topological pace I 1 δ1 and δ then the ta poduct 1 on X x Y i deined a ollow : ( 1 ) (x, y) (α) = 1(x) 1 x 44 (α) Λ (y) (α), o evey (x, y) X x Y and o evey α Ι The poduct topology δ δ on X x Y i the econd ode uzzy topology having the collection { 1 / 1 δ 1, δ } a a bai Theoem 34 Poduct o two (S H) 1 pace i (S H) 1 Poo Let (X, δ 1 ) and (Y, δ ) be two (S H) 1 pace Conide two ditinct uzzy point z, u in X x Y whee z = (x, y) and u = (p, q) Eithe x p o y q Aume x p Then x p Thee exit, g δ 1 uch that x, p g and Λ 1 g = 0

5 Second Ode Fuzzy S-Haudo Space, g δ 1 1, g 1 δ 1 x δ Fo α Ι, conide ( 1) (x, y) (α) = (x) (α) Λ 1 (y) (α), α Ι ( x ) Similaly z 1 u g 1 Alo Λ 1 g = 0 ( 1) Λ 1 ( g 1) = 0 The poo i imila i y q Hence (X x Y, δ 1 x δ ) i (S H) 1 Deinition 35 Let {(X λ, δ λ ) / λ Λ} be a amily o econd ode uzzy topological pace and X = Π X λ The poduct topology on X i the one with λ Λ baic econd ode uzzy open et o the om except o initely many λ Π λ Λ λ whee λ δ λ and λ = 1 Hee ( Π λ Λ λ ) ((xλ ) λ Λ ) (α) = Λ λ Λ λ (xλ ) (α), o evey (x λ ) λ Λ and o evey α Ι Theoem 36 Abitay poduct o (S H) 1 pace i (S H) 1 Π λ Λ X λ Theoem 37 (X, δ) i uzzy S-Haudo i (X, δ ) i (S H) 1 whee (X, δ ) i got om (X, δ) though the aociation / c 1 Poo Given δ, δ = { / δ} whee (x) (α) = (x), o evey x X and α Ι Aume (X,δ) i uzzy S-Haudo Conide two ditinct econd ode uzzy point x, y in X Then the it ode uzzy point x, y in X ae ditinct thee exit, g δ uch that x, y g and Λ g = 0, g δ and x (x) (x) (α), α Ι x Similaly y g Alo Λ g = 0 Λ 1 g = 0 Hence (X, δ ) i (S H) 1 Poo o the convee i imila Theoem 38 I (X, δ ) i (S H) 1 then o evey α Ι, (X, δ α ) i uzzy S- Haudo whee (X, δ α ) i got om (X, δ ) though the aociation c/ 3 45

6 AKalaichelvi Poo Given δ, δ α = { α / δ } whee α (x) = (x) (α), x X Conide two ditinct it ode uzzy point x, y in X Then ditinct thee exit, g δ uch that x x, y g and Λ 1 g = 0 (x) (α), α Ι α (x) x α Similaly y g α Alo Λ1 g = 0 α Λ g α = 0 (X, δ α ) i uzzy S-Haudo x, y ae Theoem 39 I (X, δ ) i (S H) 1, then (X, δ ) i uzzy S-Haudo whee (X, δ ) i got om (X, δ ) though the aociation c/ 4 Poo Given δ, a ubbae o δ i S = { / δ } whee (x) = x X Conide two ditinct it ode uzzy point, x, y in X Then ditinct Thee exit, g δ uch that x, y g and Λ 1 g = 0 x (x) (α), α Ι Similaly y g Alo Λ 1 g = 0 Λ g = 0 (X, δ ) i uzzy S-Haudo α Ι α Ι λ (x)(α), x, y (x) (α) (x) x Theoem 310 (X, δ ) i (S H) 1 i (X, ( δ ) c ) i (S H) 1 whee (X, ( δ ) c ) i got om (X, δ ) though the aociation c/ 6 Poo The poo i immediate om the ollowing two act Fo, g δ and a econd ode uzzy point x in X, (1) x x ( ) c () Λ1 g = 0 ( ) c Λ 1 ( g ) c = 0 Theoem 311 I (X, δ ) i (S H) 1, then (1) Fo (0, 1), (X, i ( δ )) i Haudo () (X, i ( δ ) ) i Haudo ae 46

7 (3) (X, i( δ )) i Haudo Second Ode Fuzzy S-Haudo Space Poo Conide x, y X uch that x y Conide, Ι uch that >, > in X ae ditinct Thee exit, g δ uch that x y, y g and Λ 1 g = 0 (A ) x (x) (α), α Ι ( (x)) 1 (, 1] = Ι x Similaly y (A ) g, g δ (A ), (A ) g Alo Λ1 g = 0 (A ) (X, i ( δ )) i Haudo Poo o () and (3) ae imila i ( δ ) (A ) g Theoem 31 I (X, ) i Haudo, then = ϕ (1) Fo (0, 1), (X, (ω ()) ) i (S H) 1 () (X, ω ()) i (S H) 1 (3) (X, ω ( )) i (S H) 1 Poo (1) Conide two ditinct econd ode uzzy point x, y in X Then x y Thee exit, V uch that x, y V and V = ϕ x (x χ ) = 1 χ (x ) (α), α Ι x χ Similaly y χ V Alo V = ϕ χ Λ 1 χ V = 0, V χ, χ V K whee K i a bae o (ω ()) ( = ( A ), V = ( A ) χ ) χ V Hence (X, (ω ()) ) i (S H) 1 Poo o () and (3) ae imila Note 313 The theoem 4 to 1 poved o the epaation axiom (S H) 1 have exact paallel o the epaation axiom (S H) alo x, REFERENCES 1 Chang, CL, Fuzzy Topological Space, J Math Anal Appl, 4 (1968)

8 AKalaichelvi Geoge J Kli and Tina A Folge, Fuzzy et, ncetainty and Inomation, Pentice-Hall o India Pivate Limited, New Delhi, Geoge J Kli and Bo Yuan, Fuzzy Set and Fuzzy Logic: Theoy and Application, Pentice Hall o India Pivate Limited, New Delhi, Kalaichelvi, A, Second ode uzzy topological pace - Ι, Acta Ciencia Indica, XXXIII M, (3) (007) Kalaichelvi, A, Second ode uzzy topological pace - ΙΙ, Intenational Jounal o Mathematical Science and Application, 1() (011) Sivatava, R, Lal, SN and Sivatava, AK, Fuzzy Haudo Topological Space, J Math Anal Appl, 81 (1981) Sivatava, R, Lal, SN and Sivatava, AK, On Fuzzy T 0 and R 0 Topological Space, J Math Anal Appl, 136 (1988) Zadeh, LA, Fuzzy Set, Inomation and Contol, 8 (1965),