S Ü Fen Ed Fak Fen Derg Sayı 23 (2004) 21-27, KONYA On the Relations Between Fuzzy Topologies and α Cut Topologies Zekeriya GÜNEY 1 Abstract: In this study, some relations have been generated between fuzzy sets and their cross-sections, and further relations have been derived between fuzzy topological spaces and α-cut topologies by using these relations. Key words and phrases: Fuzzy set, α-cut, fuzzy topological spaces, α-cut topologies Fuzzy Topolojiler ve α-kesit Topolojiler Arasındaki Baıntılar Üzerine Özet: Bu çalımada fuzzy kümeler ve bunların α-kesitleri arasında bazı baıntılar üretilmi ve bu baıntıları kullanarak, fuzzy topolojik uzaylar ve α-kesit topolojileri arasında bazı ilikiler ortaya koyulmutur. Anahtar sözcük ve deyimler: Fuzzy küme, α-kesit, fuzzy topolojik uzaylar, α-kesit topolojiler Introduction Standard symbols and deduction formulas of logistics have been used in the definitions, assumptions, and proofs that have been provided in this paper. Universal symbolic language of mathematics have been used instead of a national language, where applicable. Let (X,F) be a fuzzy topological space (f.t.s), A be a fuzzy set in space (X,F), α (0,1] =I o, A α ={x A(x) α} be a α-cut of set A and F α ={A α A F } be the family of α-cuts of the elements of F. It has been proven that family F α a basis for at least one topology on X, and some fuzzy-crisp relations have been derived by determining some sufficient conditions for X to be a topological space. Preliminaries ( I := [0,1] ) ( Y X = {f f:x Y function } ) ( f Y X ) ( A I X ) ( B I Y ) ( A I X ) : 1. ( f -1 [B] )(x) = B( f(x) ) [1] 2. ( A )(x) := sup {A(x) A A }, ( A )(x) := inf {A(x) A A }, 3. ( X,F ) f.t.s. : ( 0 F ) ( 1 F ) ( A,B F A B F ) ( A F A F ) A open : A F, A closed : \A F 1 Mula Üniversitesi Fen-Edebiyat Fakültesi Matematik Bölümü 48000- Mula
On the Relations Between Fuzzy Topologies and α Cut Topologies 4. A o = { B ( B A )(B F ) }, A F A=A o [2] 5. B base for F : ( B F ) [ A F ( B * B )( A= B * )] [3] 6. (α I ) ( A I X ) A α ={x A(x) α}, 0 α = 7. A α := {A α A A }, A α A f : (X,F ) ( Y,E ) φ, α (0,1] X, α = 0 8. f fuzzy continuous (f-c.) : ( A E f -1 [A] F ) 9. (X,F ) T o : ( x,y X ) ( α I o )( A F )[ (P x α A)( P y α A) (P x α A)( P y α B)] α, x = a 10. P I X, P(x) = : P = P a α 0,x a A I X ( P x α A : α A(x) ) Some Results For α-cuts of Fuzzy Sets A,B I X, α I, A I X 1. A B A α B α, A = B A α = B α 2. ( A B ) α = A α B α, ( A B ) α = A α B α [4] 3. A α ( A ) α, ( A ) α A α Proof: 2.2. A A ( A A ) ( A A ) 3.1 ( A α ( A) α ) ( ( A ) α A α ) A α ( A) α, 2.7 ( A α ( A) α ) ( ( A ) α A α ) 4. ( A I X ) ( α I ) ( ( A ) α A α ), A I X ) ( α I ) ( ( A ) α A α ) ( Example ) X=R, a I : A a = {(x,a) x R }, A = { A a a ( 1/2, 2/3 ) } 2 X, for α = 2/3 A 2/3 = { ( A a ) 2/3 a ( 1/2, 2/3 ) }={ φ }, A 2/3 = φ, A = A 2/3 = {(x,2/3) x R }, ( A ) 2/3 = R,... 5. A < ℵ o ( ( A) α = A α ) ( ( A ) α = A α ) Proof 2.6 2.2. x ( A) α ( A)(x) α sup {A(x) A A } α hi p. max {A(x) A A } α ( A A )( A(x) α ) 2.6. ( A A)( x A α ) x A α, 2.6 2.2. ( A) α = {x ( A)(x) α } = {x inf {A(x) A A } α } hip. = {x min {A(x) A A } α } = { x A A A(x) α } 2.6. 22
Zekeriya GÜNEY = {x A A x A α } = A α. 6. A A ( A) α = A α, A A ( A ) α = A α 2.7. 3.3. A A ( A) α A α ( A) α A α ( A) α = A α : 2.7. 3.3. A A ( A ) α A α ( A ) α A α ( A ) α = A α 7. (\A) α = \A 1-α A -1 (1-α) x (\A) α (\A)(x) α 1- A(x) α A(x) 1-α A(x) < 1-α A(x) = 1-α x A 1-α x A -1 (1-α) x \ A 1-α x A -1 (1-α) x \A 1-α A -1 (1-α). 8. (A\B) α = ( A α \ B 1-α ) ( A α B -1 (1-α) ) [4] 9. ( f Y X ) ( A I Y ) ( α I ) ( f -1 [A] ) α = f -1 [A α ] ) 2.6 2.1. x ( f -1 [A] ) α ( f -1 [A] ) (x) α A( f(x) ) α 2.6. f(x) A α x f -1 [A α ] Some Relatıons Between Fuzzy Topologies and α-cut Topologies: 1. (X,F) f.t.s. ( α I o )( (X,E ) t.s. ) ( F α base for E ) 2.3 (X,F ) f.t.s. 1 F 2.6-2.7 ( α I o ) ( 1 α = X F α )... (a) 2..7. A, B F α ( α I o ) ( C,D F )( A=C α )( B= D α ) hi p. - 2..3. 2..7. C D F ( C D ) α F α 3..2. C α D α = A B F α...(b), (a),(b) ( α I o )( (X,E ) t.s. ) ( F α base for E ). 2. ( F I X ) ( F < ℵ o ) [ (X,F) f.t.s. ( α I o )( (X,F α ) t.s. ) ] i): 2.3 (X,F ), f.t.s. ( 0 F ) ( 1 F ) 2.6-2.7 ( α I o ) ( 0 α = φ F α ) ( 1 α = X F α )...(a), 4.1 A, B F α A B F α...(b), 2. 7. A F α ( B F ) ( A = B α ) F f.t. B F 2..7 hip,-3.5. ( B ) α F α B α = A F α...(c), 23
On the Relations Between Fuzzy Topologies and α Cut Topologies (a),(b),(c) ( α I o (X, F α ) t.s. ), ii) : (X,F) not f.t.s. 2.4. ( 0 F ) ( 1 F ) ( A,B F)( A B F ) ( A F) ( A F )...(a), 0 F ( A F )( a X)( 0< A(a) 1) α =A(a)/2 ( α I o ) ( A F ) ( a X) (A(a ) > α ) ( α I o ) ( A F ) ( a A α ) ( α I o ) ( φ F α )...(b), 1 F ( A F ) ( x X)(0 A(x) <1) ( A F ) ( A 1 X ) ( α I o ) ( X F α )...(c), 3.1 ( A,B F )( A B F ) [ α I o \ {α ( C F ) ( (A B) x = C x ) ( A B C ) }] ( A α, B α F α ) ( (A B) α F α ) 3.2. ( α I o ) ( A α, B α F α ) ( A α B α F α ).. (d), 3.1. ( A F) ( A F ) K= I \ {x ( C F ) ( ( A ) x = C x ) ( A C ) } φ Example: Hip.-3.5. α K ( α I o ) ( A α F α ) ( ( A ) α F α ) ( B =A α F α ) ( B = A α F α ) (a),(b),(c),(d), (e) ( α I o ) [ (X,F α ) not t.s. ]....(e), X=(0,1), A = {(x, -x/6+2/3 ) x X }, B ={(x, x/2 +1/4 ) x X }, F = { 0, 1, A, B, A B, A B } F = 6 < ℵ o, x/2 +1/4, 0 < x 5/8 x/6+2/30, 0 < x 5/8 (A B)(x) =,(A B)(x) = x/6+2/30, 5/8 < x < 1 x/2 +1/4, 5/8 < x < 1 X, 0 < α ½ X, 0 < α 1/4 A α = (0, 4-6α ], 1/2 < α < 2/3, B α = [2α 1/2,1), 1/4 < α < 3/4 φ, 2/3 α 1 φ, 3/4 α 1 X, 0 < α 9/16 (0, 4-6α ] [2α 1/2,1), 9/16 < α < 2/3 (A B) α = A α B α = [2α 1/2,1), 2/3 α < 3/4 φ, 3/4 α 1 (A B) α = A α B α = X, 0 < α 1/4 [ 2α 1/2,1), 1/4 < α < 1/2 [ 2α 1/2, 4-6α ], 1/2 α 9/16 φ, 9/16< α 1 24
Zekeriya GÜNEY F α = { 0 α, 1 α, A α, B α, (A B) α, (A B) α }= { φ, X, A α, B α, A α B α, A α B α } { φ, X }, 0 < α 1/4 3/4 < α 1 {φ, X, [ 2α 1/2,1) }, 1/4 < α 1/2 2/3 < α < 3/4 = {φ, X, [ 2α 1/2,1), (0, 4-6α ], [ 2α 1/2, 4-6α ] }, 1/2 < α 9/16 {φ, X,[ 2α 1/2,1), (0, 4-6α ], (0, 4-6α ] [2α 1/2,1) }, 9/16< α< 2/3 ( α I o )( (X,F α ) t.s. ). 3. { A A F } F I X [ (X,F) f.t.s. ( α I o )( (X,F α ) t.s. ) ] from 3.6., 4.1. and 4.2. 4. ( (X,F) f.t.s. ) ( α I o )( (X,F α ) not t.s. ) ( Example ) 0, x < a X = R, A a (x) = x-a, a x a+1, F = { 0,1 } { A a a 0 } 1, a+1 < x (X,F) f.t.s., but for α =1/2, F α ={φ, R } { [a+1/2, ) a 0 }and A = { [a+1/2, ) a > 0 } F α, A = (1/2, ) F α and (R,F α ) not t.s. 5. ( X )( A I X )( α I o )( (X, A α ) t.s. ) ( (X, A ) not f-t.s. ) ( Example ) X=R, A b ={(x,b) x R }, A = { 0,1 } { A b 0<b<1/2 }, α I o A α = { 0,1 }, (X, A α ) t.s., but (X, A ) not f-t.s.. 6. ( (X,F) f.t.s. ) (α I o ) ( (X,F α ) t.s. ) ( A X ) ( A o ) α ( A α ) o 3.1. A o A ( A o ) α A α hip. 2.7. ( A o ) α (A α ) o A o F ( A o ) α F α 7. ( (X,F) f.t.s. ) (α I o ) ( (X,F α ) t.s. ) ( A F ) ( A o ) α = ( A α ) o Proof.: 3-1. A F A o = A (A o ) α = A α A F 2.7. hip. (A o ) α = (A α ) o A α F α (A α ) o = A α 8. ( (X,F) f.t.s. ) (Y, E ) f.t.s. ) Proof.: (α I o ) ( (X,F α ) t.s. ) (Y, E α ) t.s. ) ( f : (X,F ) ( Y,E ) f-c. ) f: (X,F α ) (Y,E α ) c. 2.7. Hip-2.10- B E α ( A E )( B = A α ) f -1 [A] F 2.7. 3.9. ( f -1 [A] ) α F α f -1 [A α ] = f -1 [B] F α f: (X,F α ) (Y,E α ) c. 25
On the Relations Between Fuzzy Topologies and α Cut Topologies 9. f : (X,F ) ( Y,E ) f-c. ( A I Y ) ( f -1 [A o ] ( f -1 [A] ) o ) Proof.: i. : Hip.. A I Y A o E f -1 [A o ] F 2.4 A o A f -1 [A o ] f -1 [A] ii. : 2.4 A E A o =A f -1 [A o ] = f -1 [A] f -1 [A o ] ( f -1 [A] ) o hip. 2.4 f -1 [A] ( f -1 [A] ) o f -1 [A] = ( f -1 [A] ) o 2.4 f -1 [A] F / f f-c. 10.. ( (X,F ) T o ) ( β I o )( (X, F β ) t.s ) (X, F β ) T o Proof.: 2.10 (X,F ) T o ( x,y X ) ( α I o )( A F )[ (P α x A)( P α y A) (P α x A)( P α y B)] 2.7. ( x,y X ) (A β F β )[ (x A β )(y A β ) (x A β )(y B β )] (X, F ) T o Example: X = { a, b }, F = { 0, 1, P a 1/3 }, A= P a 1/3 A(x) = 0, x a, for α =1/4, A α ={a}, F α = { φ, X, {a} } T o 1/3, x = a Concluding Remarks In this study, various basic theorems on fuzzy sets and their a-cross-sections, and topological fuzzy-crisp relations resulting from these theorems have been introduced. Similarly, many fuzzy-topological concepts can be related with their equivalent crisp-topological concepts. In the proofs presented here, pure symbolic and formal language of mathematics, which is a universal mean of communication, has been deliberately used instead of a national language. References: [1] Hu Cheng- Ming, Fuzzy Topological Spaces, Journal of Mathematical Analysis andapplications, 110, 141-178, ( 1985) [2] Chang, C.L., Fuzzy Topolocical Spaces, Journal of Mathematical Analysis andapplications, 24, 182-190, ( 1968 ) [3] Husain, T., Almost Continuous Mappings. Prace Math., 10, 1-7(1966) [4] Güney, Z., Fuzzy Kümelerin α- kesitlerine dair bazı sonuçlar ve bir düzeltme, Ulusal Matematik Sempozyumu, Çukurova Üniv. Adana, 1995 26