β COMPACT SPACES IN FUZZIFYING TOPOLOGY *

Transcription

1 Iraia Joural of Siee & Tehology, Trasatio A, Vol 30, No A3 Prited i The Islami Republi of Ira, 2006 Shiraz Uiversity FUZZ IRRESOLUTE FUNCTIONS AND FUZZ COMPACT SPACES IN FUZZIFING TOPOLOG * O R SAED ** AND M AZAB ABD-ALLAH Departmet of Mathematis, Faulty of Siee, Assiut Uiversity, Assiut 756, Egypt sayed_o_r@yahooom Abstrat I this paper the oepts of fuzzifyig irresolute futios ad fuzzifyig ompat spaes are haraterized i terms of fuzzifyig ope sets ad some of their properties are disussed Keywords Lukasiewiz logi, fuzzifyig topology, irresoluteess, fuzzifyig ompatess, ompatess INTRODUCTION Fuzzy topology, as a importat researh field i fuzzy set theory, has bee developed ito quite a mature disiplie [-6] I otrast to lassial topology, fuzzy topology is edowed with riher struture, to a ertai extet, whih is maifested i differet ways to geeralize ertai lassial oepts So far, aordig to [2], the kid of topologies defied by Chag [7] ad Gogue [8] are alled the topologies of fuzzy subsets, ad further, are aturally alled L-topologial spaes if a lattie L of membership values has bee hose Loosely speakig, a topology of fuzzy subsets (resp a L-topologial spae is a family τ of fuzzy subsets (resp L-fuzzy subsets of oempty set, ad τ satisfies the basi oditios of lassial topologies [9] O the other had, the authors of [0, ] proposed the termiologies I-fuzzy topologies (if the set of membership values is hose to be the uit iterval [0, ] ad L-fuzzy topologies (if the orrespodig set of membership values is hose to be lattie L I [2], a L-fuzzy topology is a L-valued mappig o the traditioal power set P( of I [4-5, 0, ] a L-fuzzy topology is a L-valued mappig o the L- valued mappig o the L-power set L of I 99, ig [3-6] used the semati method of otiuous valued logi to propose so-alled fuzzifyig topology as prelimiary researh o bifuzzy topology ad to give a elemetary developmet of topology i the theory of fuzzy sets from a ompletely differet diretio Briefly speakig, a fuzzifyig topology o a set assigs eah risp subset of to a ertai degree of beig ope, other tha beig defiitely ope or ot I fat fuzzifyig topologies are a speial ase of the L-fuzzy topologies i [0, ] sie all the t-orms o I are iluded as a speial lass of tesor produts i these paper ig uses oe partiular tesor produt, amely the Lukasiewiz ojutio Thus his fuzzifyig topologies are a speial lass of all I-fuzzy topologies osidered i the ategorial framework of [0, ] Partiularly, as the author [3-6] idiated, by ivestigatig fuzzifyig topology we may partially aswer a importat questio proposed by Rosser ad Turquette [7] i 952, whih asked whether there Reeived by the editor September 4, 2005 ad i fial revised form Jauary 24, 2007 Correspodig author

2 298 O R Sayed / M Azab Abd-allah are may valued theories beyod the level of prediates alulus Roughly speakig, the sematially aalysis approah trasforms formal statemets of iterest, whih are usually expressed as impliatio formulas i logial laguage, ito some iequalities i the truth value set by truth valuatio rules, ad the these iequalities are demostrated i a algebrai way ad the semati validity of olusios is thus established So far, there has bee sigifiat researh o fuzzifyig topologies For example, ig [6] itrodued the oept of ompatess ad established a geeralizatio of Tyhooff's theorem i the framework of fuzzifyig topology I [8] the oepts of fuzzifyig ope sets ad otiuity were itrodued ad studied Also, Sayed [9] itrodued ad studied the oept of fuzzifyig Hausdorff separatio axiom I lassial mathematis, the irresolute futio has bee give i [20] ad the oept of ompat spaes has bee defied ad some of its properties have bee obtaied i [2] I [22] the oept of fuzzy irresolute futio was haraterized ad ivestigated Also, i [23] the oept of ompatess for fuzzy topologial spaes was itrodued ad disussed I this paper we itrodue ad study the oept of the irresolute futio betwee fuzzifyig topologial spaes Furthermore, we itrodue ad study the oept of ompatess i the framework of fuzzifyig topology We use the fiite itersetio property to give a haraterizatio of the fuzzifyig ompat spaes 2 PRELIMINARIES I this setio, we offer some oepts ad results i fuzzifyig topology whih will be used i the sequel For the details, we refer to [8, 3-6] First, we display the Lukasiewiz logi ad orrespodig set theoretial otatios used i this paper For ay formula ϕ, the symbol ϕ meas the truth value of ϕ, where the set of truth values is the uit iterval [0, ] We write = ϕ if ϕ = for ay iterpretatio By = w ϕ( ϕ is feebly valid we mea that for ay valuatio it always holds that ϕ > 0, ws ad ϕ = ψwe mea that ϕ > 0 implies ψ = The truth valuatio rules for primary fuzzy logial formulae ad orrespodig set theoretial otatios are: ( (a α = α( α [0,] ; (b ϕ ψ mi ( ϕ, ψ = ; ( ϕ ψ mi(, ϕ ψ = + (2 If A I(, x A : = A ( x (3 If is the uiverse of disourse, the xϕ( x : = if ϕ( x x I additio, the truth valuatio rules for some derived formulae are ( ϕ : [ ϕ 0] ϕ = = ; (2 ϕ ψ : ( ϕ ψ max ( ϕ, ψ = = ; (3 ϕ ψ : ( ϕ ψ ( ψ ϕ = ; (4 [ ϕ ψ]: = ( ϕ ψ = max( 0, ϕ + ψ ; (5 ϕ ψ = ϕ ψ = ( ϕ + ψ [ ]: mi, ; (6 xϕ( x : = x ϕ( x : = sup ϕ(; x (7 If AB, I(, the x (a A B = x( x A x B = ( A x + B x : if mi, ( ( ; x Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006

3 Fuzzy irresolute futios ad 299 (b A B : A B B A = ; ( A B where I( is the family of all fuzzy sets i Ofte we do ot distiguish the oetives ad their truth value futios, but stritly state our results o formalizatio as ig [3-6] did Seod, we give some defiitios ad results i fuzzifyig topology Defiitio 2 [3] Let be a uiverse of disourse, τ I ( P (, satisfyig the followig oditios: ( τ( =, τ( φ = ; (2 for ay ( AB,, τ A B τ( A τ( B; λ:, λ λ λ Λ λ Λ (3 for ay { A λ Λ } τ A τ( A The τ is alled a fuzzifyig topology ad(, τ is a fuzzifyig topologial spae Defiitio 2 2 [3] The family of all fuzzifyig losed sets, deoted by F I( P as A F: = A τ, where A is the omplemet of A (, is defied Defiitio 2 3 [3] The fuzzifyig eighborhood system of a poit x by Nx ( P( I ad defied as follows: N ( A = sup τ ( B x x B A is deoted Defiitio 2 4 [3, Lemma 5 2] The losure A of A is defied as Ax ( = N ( A x I Theorem 53 [20], ig proved that the losure : P ( I( is a fuzzifyig losure operator (see Defiitio 53 [3] beause its extesio : { x: A( x α } I( I (, A = α A, A I(, where α [0,] A = is the α ut of A ad αax ( = α Ax ( satisfies the followig Kuratowski losure α axioms: ( = φ= φ; (2 for ay A I (, = A A ; (3 for ay AB, I (, = A B A B; A, B I (, = A A, where( A B x = ( A x B x (4 for ay ( ( max (, ( α Defiitio 2 5 [4] For ay A, the fuzzy set of the iterior poits of A is alled the iterior of A, ad give as follows: A ( x: = N ( A From Lemma 3 [3] ad the defiitios of N ( A ad A we have τ( A = if A ( x x A x (,, where A is the omplemet of A Defiitio 2 6 [8] For ay A I = ( A ( A ad ( A ( x = A ( x Defiitio 2 7 [4, Lemma 5] If (, τ is a fuzzifyig topologial spae,, whih is give as x the τ I ( P (, Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

4 300 O R Sayed / M Azab Abd-allah ( τ V τ : = ( U ( U ( V = U, ie, τ ( V = sup τ( U is a fuzzifyig topology o ad is V = U alled the relative fuzzifyig topology of τ with respet to If, σ= τ, the (, σ is alled a subspae of (, τ Defiitio 2 8 [4, Theorem 5] Let (, σ be a subspae of (, τ For ay A, A F : = ( F ( ( F F ( A= F, where F ad F are fuzzy families of τ, σ losed sets i ad, respetively Lemma 2 [8] If A B =, the ( = A B (2 = ( A ( B Defiitio 2 9 [8] Let (, τ be a fuzzifyig topologial spae ( The family of all fuzzifyig ope sets, deoted by τ I( P (, is defied as follows: A τ : = x( x A x A, ie, τ ( A = if A ( x x A (2 The family of all fuzzifyig losed sets, deoted by F I ( P(, is defied as follows: A F : = A τ (3 The fuzzifyig eighborhood system of a poit x is deoted by Nx I( P( ad defied as follows: Nx ( A = sup τ ( B x B A (4 The fuzzifyig losure of a set A P(, deoted by l I(, is defied as follows: l ( A( x = Nx ( A (5 Let (, τ ad (, σ be two fuzzifyig topologial spaes ad let f A uary fuzzy prediate C I(, alled fuzzifyig otiuity, is give as follows: ( σ τ C ( f : = B B f ( B Defiitio 20 [9] Let Ω be the lass of all fuzzifyig topologial spaes The uary fuzzy prediate T 2 (fuzzifyig Hausdorff I( Ω is defied as follows: ( ( ( x y φ T2 (, τ : = x y x y x=/ y B C B N C N B C = Defiitio 2 [6] Let be a set If A I( of A is fiite, the A is said to be fiite ad we write ( I I(, alled fuzzy fiiteess, is give as FF( A ; = ( B ( F( B ( A B = if { α [0,]: F( A α } = if { α [ 0,]: F( A [ α] }, A = x : A ( x α ad A = { x : A ( x > α [ ] where α { } α suh that the support supp A= { x : A( x > 0 } F A A uary fuzzy prediate FF ( } Defiitio 2 2 [6] Let be a set K I I ( P( P(, alled fuzzifyig overig, is give as follows: ( A biary fuzzy prediate ( K( R, A: = x( x A B( B R x B (2 Let (, τ be a fuzzifyig topologial spae A biary fuzzy prediate K I( I ( P P fuzzifyig ope overig, is give as follows: K ( R, A: = K( R, A ( R τ Defiitio 2 3 [6] Let Ω be the lass of all fuzzifyig topologial spaes A uary fuzzy prediate Γ I( Ω, alled fuzzifyig ompatess, is give as follows: ( ( ( (, τ Γ : = R K ( R, (( R K(, A FF(, where R meas that for ay Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006 ( (, alled

5 Fuzzy irresolute futios ad 30 M P(, ( M R ( M ( Defiitio 2 4 [6] Let be a set A uary fuzzy prediate fi I I ( P( itersetio property, is give as follows: ( ( fi ( R : = ( R FF( ( x ( B(( B ( x B Lemma 2 2 [9] Let (, τ be a fuzzifyig topologial spae The ( = τ τ ; (2 = F F, (3 F A λ F ( Λ Aλ λ Λ λ Λ Corollary 2 [8] ( A = if N ( A τ x A Theorem 2 [8] For ay x A B = A B ( A Nx B Nx x,,, 3 IRRESOLUTE FUNCTIONS, alled fuzzifyig fiite Defiitio 3 Let (, τ ad (, σ be two fuzzifyig topologial spaes ad let f A uary fuzzy prediate I I(, alled fuzzifyig irresolute, is give as I ( f : = B B σ f ( B τ follows: ( Theorem 3 Let (, τ ad (, σ be two fuzzifyig topologial spaes ad let f The = f I f C Proof: From Lemma 22 we have σ( B σ ( B ad the result holds Defiitio 3 2 Let (, τ ad (, σ be two fuzzifyig topologial spaes ad let f We defie the uary fuzzy prediates w I(, where k =, 5, as follows: ( f w B( B F f B F k = (, where F ad F are the fuzzifyig losed subsets of ad, respetively (2 f w x u( u N f u Nx 2 = ( (, where f x eighborhood systems of ad, respetively; (3 f w x u( u N = f x v( f v u v Nx (4 f w4 = A( f ( l ( A l ( f( A ; (5 f w5 = B( l ( f ( B f ( l ( B 3 ( ( ; Theorem 3 2 = f I f w, k=,,5 Proof: (a We will prove that = f I f w k N ad N ( ( f w = B F ( B F f ( B ( σ ( τ ( ( = B B f B ( σ ( τ ( ( = B B f B are the family of fuzzifyig Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

6 302 O R Sayed / M Azab Abd-allah ( σ ( τ ( ( = u u f u = f (b First, we prove that f I f w 2 employig the rules of Lukasiewiz logi ad the lear fat that fx ( A uimplies x f ( A f ( u : Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006 I ( σ τ ( f I : = A ( A f ( A ( ( ( σ( τ ( ( B f x A u A f A ( ( σ( ( ( τ ( ( A f x A u A A f x A u f A ( σ ( τ A f( x A u ( A A x B f ( u ( B From this, the required f I f w 2 is followed by the rule of geeralizatio (o x ad u i Lukasiewiz logi Seod, we prove that f w 2 f I, by the rules of Lukasiewiz logi ad employig Corollary 2: ( x ( f w 2 = u x Nf( x ( u N f ( u ( ( ( f( x ( ( x ( u x f u N u x f u N f u ( σ τ ( = u ( u f ( u f I = ( We prove that f w 2 f w = 3 From Theorem 2 we have f w 3 if if mi, N ( f( x ( u sup N x ( v = + x u P v P(, f( v u if if mi, N ( ( sup ( ( 2 ( f x u N x f u f w x u P + = v P(, f( v u (d We prove that f w 4 f w = 5 First, sie for ay fuzzy set A we have f ( f ( A A =, the ( ( ( f f l f ( B l ( f ( B = for ay B P ( Also, sie f ( f (( B B =, the ( ( l f f ( B l ( B = Therefore, from Lemma 2 (2 [22] we have Therefore ( ( ( ( ( ( ( ( ( ( l f B f l B f f l f B f l B ( ( ( ( ( ( ( ( f f l f B f l f f B ( f l f ( B l f f ( B ( ( ( (( ( ( f w 5 = B l f ( B f l ( B

7 Fuzzy irresolute futios ad 303 ( ( ( ( ( ( ( B f l f B l f f B ( ( ( 4 A f l ( A l f ( A = f w Seod, for eah A, there exists B suh that fa ( = Bad f ( B A From Lemma 2 ( [5] we have (e We wat to prove that = f w2 f w5 ( ( ( f w 4 = A f l ( A l f ( A ( ( ( ( ( ( ( A f l A f f l f A ( ( ( ( ( Al A f l fa ( ( ( ( B= f ( A l f ( B f l B ( ( ( B l f ( B f l ( B f w = 5 B P( x ( ( f w5 = if l f ( B f l ( B B P( ( ( ( Nx f B N B = if if, ( + ( B P( x f( x ( N ( ( f( x B Nx f B = if if mi, + ( B P( x ( x ( = if if mi, Nf( x ( u + N f ( u = f w 2 4 COMPACTNESS IN FUZZIFING TOPOLOG Defiitio 4 A fuzzifyig topologial spae (, τ if τ ( A B τ ( A τ ( B Defiitio 42 A biary fuzzy prediate K ( P P is give as K ( R, A: = K( R, A ( R τ is said to be fuzzifyig topologial spae I I ( ( (, alled fuzzifyig ope overig, Defiitio 43 Let Ω be the lass of all fuzzifyig topologial spaes A uary fuzzy prediate Γ I( Ω, alled fuzzifyig ompatess, is give as follows: ( (, τ Γ : = ( R ( K ( R, ( (( R K(, FF( (2 If A, the Γ A = Γ ( A τ A ( :, Lemma 4 = K ( R, A K ( R, A Proof: Sie from Lemma 22 = τ τ, the we have R τ R τ K ( R, A K( R, A So, Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

8 304 O R Sayed / M Azab Abd-allah Theorem 4 = (, τ Γ (, τ Γ Proof: From Lemma 4 the proof is immediate Theorem 4 2 For ay fuzzifyig topologial spae (, τ ad A, Γ ( A ( R K ( R, A ( ( R K(, A FF(, K is related to τ ( ( Proof: For ay R I( I(, we set R I( I( A defied as ( C ( B K( R A = R C = R B = R B = K( R A R = R with C = A B, B The, if sup ( if sup ( if sup (,, beause x A ad x B if x A x C x A x C = A B x A x B ad oly if x A B Therefore ( τ τ A R = if mi, R ( C + A( C C A = if mi, sup ( B sup τ ( B C A R + C = A B, B C = A B, B C A, C = A B, B ( B τ B sup mi, R ( + ( ( B τ B if mi, R ( + ( = τ R B For ay R, we defie I ( P ( as follows: The,FF( FF( Furthermore, we have R = ad ( The ( where K ( R, A = K ( R, A ( R τ A ( B if B A, ( B = 0 otherwise K, A = K(, A ( Γ ( A K ( R, A Γ ( A K R, A ( K, A FF ( ( ( ( R ( ( ( K (, A FF ( R ( K, A FF ( ( ( ( Β Β R Β Β ( Γ ( A K ( R, A ( Β Β R K( Β, A FF( Β, Therefore Coversely, for ay ( PA (, ( ( P ( ( Γ ( A if K ( R, A ( Β Β R K( Β, A FF( Β R I ( K A K A FF (( = ( R ( R, ( Β Β R ( Β, ( Β R I if R τ A = if mi(, R ( B + τ A( B B A Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006

9 Fuzzy irresolute futios ad = λ, the for ay N ad B A, sup τ( C = τ A ( B > λ+r( B, B= A C, C exists CB suh that CB A= B ad τ ( CB > λ+r( B Now, we defie R I( P ( as R ( C = max( 0, λ +R( B The R τ B A = ad R = R = R if sup ( λ + R( (, if sup ( if sup ( B K A C C x A x C x Ax B B x A x B = if sup R ( B + λ = K( R, A + λ, x A x B ( R, = ( R, ( R τ K( A K( A K A K A ( λ = R, max 0, R, + max ( 0, K( R, A + λ = K (, A R For ay R, we set I ( PA ( as ( B = ( C, B A The R,FF( = FF( ad K(, A = K(, A Therefore ( B (( ( ( R K ( R, A ( R K, A FF( K ( R, A ( ( K (, A ( (( K(, A FF( ( K ( R, A R R R ( ( ( ( ( K R, A ( R K, A FF( K R, A ad 305 there ( (( K(, A FF( ( ( ( R Theorem 4 3 Let (, τ ( K A FF ( ( ( ( K, A FF ( R, ( Β Β R Β Β Let We obtai ( K ( A ( ( K( A FF( ( (,, R R R ( (( K (, A FF ( Β Β R Β Β The ( K ( A ( (( K FF ( R R, R (, R ( (( ( ( K ( R, A Β Β R K( Β, A FF( Β ( ( ( ( if K ( R, A Β Β R K( Β, A FF( Β R I P ( be a fuzzifyig topologial spae K ( R, A = Γ ( A ( ( ( ( ( π : = ( R R I( P ( R F fi( R x A A R x A; π2 : = ( R ( B ((( R F ( B τ ( (( R FF ( ( B ( R B The = Γ (, τ πi, i=,2 Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

10 306 O R Sayed / M Azab Abd-allah Proof: (a We prove Γ (, τ = π For ay R I( P (, we set R ( ( A =R A The A P( A P( ( A τ A τ R = if mi, R ( + ( ( = if mi, R ( A + F ( A = F R, FF( R = if { α [0,]: F( R α } = if { α [0,]: F( R α } = FF( R ad Β R Β( M R ( M Β ( M R( M Β R Therefore, ( ( ( ( ( Γ (, τ = R K ( R, K, FF( ( K (, ( K, FF ( ( ( τ ( ( ( = R R R R ( ( K(, ( K(, FF( ( ( τ ( ( = R R R R = R R ( ( ( ( F (( x ( A( A R x A ( ( K (, FF ( ( ( ( F (( x ( A( A R x A = R R ( R ( Β ( Β R K( Β, FF( Β ( ( F (( x ( A( A R x A = (b We prove [ ] [ ] π 2 = R R ( ( ( FF( ( x ( B( B x B Β Β R Β Β = ( R (( R F ( (( Β ( Β R FF( Β ( x ( B( B Β x B ( x ( A( A R x A = [( R ( R F ( fi ( R ( ( x( A( A R x A ] ( R ( R F fi ( R ( x( x( A R x A ( ] [ ( ] = [ π ] π = Let B P( = if A P( mi For ay I( P( R, [( R F ( B τ ] = [ R F ( B F ] = if mi (,R( A + F ( A F ( B A P( Therefore, for ay I( P(, (,R( A + F ( A if mi(,[ A { B} ] F ( A + A P( (,[ R { B} ( A ] + F ( = if mi A P = [( R { B } ] A ( F Β let = Β \{ B } I( P( Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006

11 Fuzzy irresolute futios ad 307 Β ( A, A=/ B ( A = 0, A = B The Β, { B } Β, FF( = FF( Β, R = Β ( R { B } ad ( ( ( R FF ( ( x ( A ( A ( { B } ( x A ( ( ( { } = if mi, FF sup if B ( A Ax ( R + x A P ( if mi, FF( sup if ( ( A Ax ( fi( { B} Β ( R { B} + Β = R x A P ( Furthermore, we have ( (( ( τ ( ( ( ( π F B FF B R R ( { } ( ( ( = π R B F R FF ( ( ( ( { } ( x A A B x A ( { } ( { } = π B F fi B R R ( x ( A ( A ( { B} x A R ( B = R Therefore π sup ( ( τ ( (( ( ( R R if F B FF R I P ( B ( B ( B R = π2 Coversely, π ( ( π (( { } { } R R = R (( R { } fi B B 2 F fi 2 \ B B F \{ } 2 R F B ( R FF( = π ( ( τ ( ( ( ( ( ( { } x A A B x A ( ( ( τ ( ( ( ( = π2 R F B R FF B ( B ( x( A (( A ( { B} ( x A R = R ( x( A ( A ( x A = R Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

12 308 O R Sayed / M Azab Abd-allah Therefore ( F fi( ( x( A ( A ( x A ( P π2 if R R R = π R I ( 5 SOME PROPERTIES OF FUZZIFING COMPACTNESS Theorem 5 For ay fuzzifyig topologial spae (, τ ad A, τ = Γ (, A F Γ ( A Proof: For ay R I( PA (, we defie R I( P ( as follows: { } The ( if α [0,]: ( R ( B if B A, R ( B = 0 otherwise { } ( FF R = F R α = if α [0,]:F( R α = FF R ad sup if ( R ( B = sup if ( R( B if ( R( B x B x B A x B A x x If x A, the for ay x A we have = sup if ( R( B sup if ( R( B x B A x B A x x = sup if ( R( B x B A x = sup if ( ( B sup R if ( R( B x A x B A x A x B A ( R B = ( R B ( R B if ( if ( if ( x B A B A x B A Therefore, ( R B = ( R B sup if ( sup if (, x x B A x AB A ( R = ( Β (( Β R ( Β ( ( (( R ( fi F F x B B x B = if mi, FF( sup if ( ( B Β R Β + R x x B = if mi, FF( sup if ( ( B fi( Β R Β + R = R x x B A We wat to prove that F ( A R F A R F I fat, from Lemma 22 (3 we have F ( A F A max 0, F ( A if mi(, ( B F A( B R = + R + B A ( B F A F A B if R ( + ( + ( B A Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006 =/

13 Fuzzy irresolute futios ad 309 ( B ( F A F A B if R ( + ( ( B A if ( ( ( sup ( R B + F A F B B A B B= B, B ( B ( F A F ( B if R ( + sup ( B A B A= B, B ( B ( F ( A B if R ( + sup B A B A= B, B ( if R ( B + F ( B B A ( B F B = if mi, R ( + ( B A ( = if mi, R ( B + F ( B = F R B A Furthermore, from Theorem 43 we have Γ (, τ F ( A F A fi( R R Γ (, R F fi( R τ ( B ( B sup if R ( = sup if R ( x x B A x x B A The Γ (, τ F ( A F R A fi R sup if R( B ( ( x A x B A if F A fi( sup if ( ( B R I( PA ( R R R x Ax B A = Γ ( A Theorem 5 2 Let (, τ ad (, σ be ay two fuzzifyig topologial spaes ad f The = Γ (, C ( f Γ ( f( τ is surjetio Proof: For ay Β I ( P (, we defie as follows: R ( A = f ( Β ( A = Β ( f( A The = if ( R, = if sup R ( = if sup Β ( ( K A f A sup Β( B = x x A x x A if sup Β( B = K( Β, f (, x f ( x B y f ( x y B [ Β σ ] [ C ( f ] = if mi (, Β( B + σ ( B if mi(,σ ( B + τ ( f ( B B B (, Β( B + ( B = max 0, if mi + B if max B σ if mi ( 0,mi(,σ ( B + ( f ( B B ( 0, mi (, Β( B + σ ( B + mi (,σ ( B + ( f ( B τ τ Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

14 30 O R Sayed / M Azab Abd-allah if mi B (, Β( B + τ ( f ( B (, Β( B + τ ( f ( if if mi B A f if ( B = A (, Β( B ( if mi +τ A A f ( B = A For ay R, if = if A A we set I( P( mi, mi sup f ( B = A Β( B + τ ( A (,R( A + τ ( A = [ R τ ] defied as follows: ( f ( A = f ( ( f ( A = ( A, A The ( f ( A = f ( ( f ( A f ( R f (( A = f f ( Β (( f ( A Β( f ( A { α [0,]: F( α } = if { α [0,] F ( f ( [ α ] } ( = if [ ] F F : K ( f ( F ( = F F F ad (, f ( = if sup ( B = if sup ( A if sup y f ( f ( y A y f ( y B ( A = if x y f ( y B = f ( A sup ( A = K x A (,, Furthermore, = [ Γ (, ] [ C ( f ] [ K τ ( Β, f ( ] [ Γ (, τ ] [ C ( f ] K( Β, f ( = [ Γ (, τ ] [ R τ ] [ K( R, ] = [ Γ (, τ ] [ K ( R, ] [( ( R K (, F F( ] [ ] [ Β σ ] [( ( R K (, f ( F F( ] [( ( R K(, f ( F F( ], where K is related to σ Therefore from Theorem 42 we obtai [ Γ (, τ ] [ C ( f ] K ( Β, f ( ( ( R K(, f ( F F( if ( K ( Β, f ( ( ( R K(, f ( F F( Β I ( P( Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006

15 Fuzzy irresolute futios ad = [ Γ( f ( ] Theorem 5 3 Let (, τ ad (, σ be ay two fuzzifyig topologial spaes ad f The = Γ (, I ( f Γ ( f( τ 3 is surjetio Proof: From the proof of Theorem 52 we have ay Β I( P ( we defie R I( P ( as follows: R ( A = f ( Β = Β( f( A The K( R, = K( Β, f( ad Β σ I ( f R τ For ay R, we set I( P ( defied as follows: ( fa ( = f( ( fa ( = ( A, A ad we have FF( FF(, ( ( K, f( K, Therefore ( ( ( K( f FF(, (, Therefore from Theorem 4 2 we obtai ( Γ (, τ I ( f K Β, f( ( = (, τ I ( f K, f( σ Γ Β Β ( Γ (, τ R τ K R, ( = Γ (, τ K R, ( ( ( K (, FF ( R ( ( ( K (, f ( FF ( R where K is related to σ Γ (, τ I( f ( (, ( ( ( (, ( ( K Β f Β K f FF ( ( K ( f ( ( K( f FF( Β I ( P ( if Β, ( Β, ( = Γ ( f ( Theorem 5 4 Let (, τ be ay fuzzifyig topologial spae ad AB, The ws ( T (, τ ( Γ ( A Γ ( B A B= φ = T (, τ 2 2 ( ; ( U ( V ( U τ ( V τ ( A U ( B V ( U V = φ ws (2 T2 (, τ Γ ( A = T2 (, τ A F Proof: ( Assume A B= φ ad T 2 (, τ = t Let x A The for ay y B ad λ < t, Corollary 2 that we have from Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

16 32 O R Sayed / M Azab Abd-allah sup { τ ( P τ ( Q : x P, y Q, P Q= φ } = sup { τ ( P τ ( Q : x P U, y Q V, U V = φ } = sup sup τ ( P sup τ Q : = sup N ( U N ( V U V = φ x P U y Q V U V = φ { x y } 2 ( { x y } if sup N ( U N ( V = T (, τ = t> λ, x y U V =/ = φ ie, there exist P, Q suh that x P, y Q, P Q = φ ad τ ( P λ, y y y y y y Β ( Qy = τp ( Qy for y B Sie τ Β =, we have ( ( ( y P ( y K, B K, B Β = Β = if sup Β( C if Β Q = if τ Q λ y B y C y B y B ( Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006 P y τ > ( Q p y > λ Set O the other had, sie T 2 (, τ ( Γ ( A Γ ( B > 0, the - t < Γ( A Γ( B Γ( A Therefore, for ay ( A t λ Γ (,, it holds that { } ( ( ( Β λ <Γ( A K Β, B + sup K, B FF λ + sup { K(, B FF( }, ie, K( B FF( Β Β { } sup, > 0 ad there exist Β suh that K(, B + FF( > 0, ie, FF( < K(, B The, if { : F( θ } K(, B there exist θ suh that θ < K(, B ad F ( θ Sie Β, we may write θ { Qy Qy } put Ux = { Py P y }, Vx { Qy Q y } Vx B, Ux Vx, Ux Py P y θ < Now, =,, We = ad have = φτ ( τ ( τ ( > λ beause (, τ is fuzzifyig topologial spae Also, τ ( Vx τ ( Qy τ ( Q y > λ I fat, if sup ( D = K(, B > θ, ad for ay y B, there exists D suh that y D ad ( D > θ, D θ y B y D Similarly, if λ ( ( A ( B Γ Γ, t, the we a fid x,, xm A with U = Ux U x m A By puttig V = Vx V xm we obtai V B, U V = φ ad ( U ( V (( U τ ( V τ ( A U ( B V ( U V φ = τ ( U τ ( V ( Ux ( Vx mi τ mi τ > λ i i=,, i=,2,, Fially, let λ t ad omplete the proof (2 Assume T 2 (, τ ( A Γ > 0 For ay x A we have from ( ( 2 sup τ ( U sup τ( U τ( V: x U, A V, U V = φ T (, τ x U A From Corollary 2 we obtai F( A = if Nx ( A = if sup τ( U T2 (, τ x A x A x U A Defiitio 5 Let (, τ ad (, σ be two fuzzifyig topologial spaes A uary fuzzy prediate i

17 Fuzzy irresolute futios ad 33 Q I (, alled fuzzifyig losedess, is give as follows: Q ( f : = B( B F f ( B F, where F ad F are fuzzy families of τ, σlosed i ad respetively Theorem 5 5 Let (, τ be a fuzzifyig topologial spae ad (, σ be a fuzzifyig topologial spae ad f The τ 2 σ = Γ (, T (, I ( f Q ( f Proof: For ay A, we have the followig: (i From Theorem 5 we have (, τ F ( A Γ Γ( A; (ii I ( F = if mi, σ ( U + τ ( f ( U ( ( A A A U P( U P( ( σ U τ A ( A f U = if mi, ( + ( = if mi, σ( U sup τ ( B U P( + A f ( U = B A U P( ( σ τ ( ( = if mi, ( U + f ( U = I ( f (iii From Theorem 53, we have ( A I ( f Γ A Γ ( f( A ws (iv From Theorem 54 (2 we have T (, σ Γ ( f( A = T (, σ F fa (, whih implies = T (, Γ ( f( A f( A F By ombiig (i-(iv we have σ Γ(, τ T (, I ( f 2 σ ( 2 F ( A Γ( A I ( f A T (, σ ( F ( A ( ( A I ( f A T 2 (, σ Γ F ( A ( f( A T 2 (, σ Γ Therefore (, 2 (, ( Γ τ T τ I f F ( A F ( f( A if ( F ( A F ( f ( A = Q B ( f A REFERENCES Hohle, U (200 May Valued Topology ad its Appliatios Kluwer Aademi Publishers, Dordreht 2 Hohle, U & Rodabaugh, S E (999 Mathematis of Fuzzy Sets: Logi, Topology, ad Measure Theory, i: Hadbook of Fuzzy Sets Series Kluwer Aademi Publishers, Dordreht 3 3 Hohle, U, Rodabaugh, S E & Sostak, A (995 Speial Issue o Fuzzy Topology Fuzzy Sets ad Systems, 73, Kubiak, T (985 O Fuzzy Topologies, Ph D Thesis, Adam Mikiewiz Uiversity, Poza, Polad 5 Liu, M & Luo, M K (998 Fuzzy Topology Sigapore, World Sietifi Autum 2006 Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3

18 34 O R Sayed / M Azab Abd-allah 6 Wag, G J (988 Theory of L-Fuzzy Topologial Spaes Shaxi Normal Uiversity Press, i' a, (i Chiese 7 Chag, C L (968 Fuzzy topologial spaes J Math Aal Appl, 24, Gogue, J A (973 The fuzzy Tyhooff Theorem J Math Aal Appl, 43, Kelley, J L (955 Geeral Topology New ork, Va Nostrad 0 Hohle, U & Sostak A (999 Axiomati foudatios of fixed-basis fuzzy topology i: Hohle U & Rodabaugh S E (999 Mathematis of Fuzzy Sets: Logi, Topology, ad Measure Theory, i: Hadbook of Fuzzy Sets Series Kluwer Aademi Publishers, Dordreht, 3, Rodabaugh, S E (999 Categorial foudatio of variable-basis fuzzy topology, i: Hohle U & Rodabaugh S E (999 (Eds, Mathematis of Fuzzy Sets: Logi, Topology, ad Measure Theory, i: Hadbook of Fuzzy Sets series Kluwer Aademi Publishers, Dordreht, 3, Hohle, U (980 Uppersemiotiuous fuzzy sets ad appliatios J Math Aal Appl, 78, ig, M S (99 A ew approah for fuzzy topology (I Fuzzy Sets ad Systems, 39, ig, M S M (992 A ew approah for fuzzy topology (II Fuzzy Sets ad Systems, 47, ig, M S (993 A ew approah for fuzzy topology (III Fuzzy Sets ad Systems, 55, ig, M S (993 Compatess i fuzzifyig topology Fuzzy Sets ad Systems, 55, Rosser, J B & Turquette, A R (952 May-Valued Logis Amsterdam, North-Hollad 8 Abd El-Hakeim, K M, Zeyada, F M & Sayed, O R (999 otiuity ad D(, otiuity i fuzzifyig topology J Fuzzy Math, 7(3, Sayed, O R (2002 O Fuzzifyig Topologial Spaes, Ph D Thesis, Assiut Uiversity, Egypt 20 Mahmoud, R A & Abd-El-Mosef, M E (990 irresolute ad topologial ivariat, Pro Pakista Aad Si ( Allam, A A & Abd-Hakeim, K M (989 O ompat spaes Bull Calutta Math So, 8, Haafy, I M & Al-Saadi, H S (200 Strog forms of otiuity i fuzzy topologial spaes Kyugpook Math J, 4, Balasubramaia, G (997 O fuzzy ompat spaes ad fuzzy extremally disoeted spaes Kyberetika, 33, Iraia Joural of Siee & Tehology, Tras A, Volume 30, Number A3 Autum 2006