Research Article Relative Smooth Topological Spaces

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1 Advances n Fuzzy Systems Volume 2009, Artcle ID , 5 pages do: /2009/ Research Artcle Relatve Smooth Topologcal Spaces B. Ghazanfar Department of Mathematcs, Faculty of Scence, Lorestan Unversty, P.O. Box 465, Khoramabad , Iran Correspondence should be addressed to B. Ghazanfar, bahman ghazanfar@yahoo.com Receved 22 June 2009; Accepted 10 November 2009 Recommended by Zne-Jung Lee In 1992, Ramadan ntroduced the concept of a smooth topologcal space and relatveness between smooth topologcal space and fuzzy topologcal space n Chang s 1968 vew ponts. In ths paper we gve a new defnton of smooth topologcal space. Ths defnton can be consdered as a generalzaton of the smooth topologcal space whch was gven by Ramadan. Some general propertes such as relatve smooth contnuty and relatve smooth compactness are studed. Copyrght 2009 B. Ghazanfar. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. 1. Introducton Let X be a nonempty set and let L, L be two lattce whch wll be copes of [0, 1] or 0, 1. The famly of all fuzzy sets on X wll be denoted by L X Zadeh [1]. In the consderaton of the nature an observer can be modeled by an operator whch evaluates each proposton by a number n the closed nterval [0, 1]; see Anvar and Molae [2] and Molae [3]. We assume that μ as a functon from X to L s an observer of X on lattce L and denote [μ] = λ L X : λ μ, whereλ μ mples that λx μxforallx X. Defnton 1.1. Let μ L X. A relatve smooth topologcal space or μ-smooth topologcal space or μ-sts for short s a trple X, μ,, where :[μ] L s a mappng satsfyng the followng propertes: μ = χ φ = 1, where χ s the characterstc functon; f λ 1, λ 2 [μ], then λ 1 λ 2 λ 1 λ 2, where s the mnmum operator n L ; λ : I λ : I. We call a smooth topology from vew pont of μ or a μ-smooth topology or a fuzzy famly of μ-open sets on X. Remark 1.2. If μ = χ X then the χ X -STS X, χ X, T χx concdes wth the smooth topologcal space X, τ defnedby Ramadan [4], and f we take L = [0, 1], L =0, 1, and μ = χ X then the χ X -STS concdes wth the known defnton of fuzzy topologcal space X, τ defned by Chang [5]. If L = L =0, 1,andμ = χ X then T χx s a classcal topology. Defnton 1.3. Let μ L X.Aμ-smooth cotopologcal space s a trple X, μ, F μ, where F μ :[μ] L s a mappng satsfyng the followng propertes: F μ μ = F μ χ φ = 1; f η 1, η 2 [μ], then F μ η 1 η 2 F μ η 1 F μ η 2 ; F μ η : I F μ η : I. We call F μ a μ-smooth co-topology or a fuzzy famly of μ-closed sets on X. Theorem 1.4. Let X, μ, be a μ-sts and F μ :[μ] L be a mappng defned by F μ η = η,whereη = μ η. Then F μ s a fuzzy famly of μ-closed sets. It s clear. It flows from η1 η 2 = μ η1 η 2 = μ sup η1, η 2 = nf μ η 1, μ η 2 = η 1 η 2. So, F μ η 1 η 2 = η 1 η 2 = η 1 η 2. 1

2 2 Advances n Fuzzy Systems It flows from I η = μ I η = μ nf η : I = sup μ η : I = I η. So, F μ η : I = η : I. Theorem 1.5. Let F μ be a fuzzy famly of μ-closed sets and defne T Fμ :[μ] L by T Fμ η = F μ η. Then T Fμ s a μ-sts on X. 2 For Γ, λj : j Γ = I T λj : j Γ μ I = T μ λj : j Γ λj : j Γ I T μ = Tμ λj : j Γ. 8 The proof s smlar to the prevous theorem. Corollary 1.6. Let be a μ-sts and F μ a fuzzy famly of μ-closed sets. Then T FTμ = and F TFμ =F μ. Suppose λ, η [μ] then we have T FTμ λ = F Tμ λ = λandf TFμ η = T Fμ η = F μ η. Example 1.7. Let X be the set ofall dfferentable real-valued functons on 1,, wth postve dervatve of order one and let L be the set of real-valued functons defned on 1,. Let μ : X L be defned by μ f = f + Exp, where Exp s the exponental functon. For nonnegatve nteger n defne λ n : X L by λ n f x = f x + n =1 x 1 1!. 3 If we take L = [0, 1] and defne : [μ] [0, 1] by χ = μ = 1; λ n = 1 1/n for n = 1, 2,... Then X, μ, saμ-sts. Snce λ n λ m = λ m, λ n λ m = λ n, where n>mand λ n = μ whenever n tends to +,so λ n λ m = λ m λ n λ m, 4 and for I N we fnd λ n : n I Tμ λ n : n I. 5 Defnton 1.8. Let Tμ 1 and Tμ 2 be two μ-smooth topologcal spaces on X. We say that Tμ 1 s fner than Tμ 2 or Tμ 2 s coarser than Tμ 1 and denoted by Tμ 1 Tμ 2 f Tμ 1 λ Tμ 2 λforevery λ [μ]. Theorem 1.9. Let Tμ : I be a famly of μ-sts on X. Then = I Tμ s also μ-sts on X,where I Tμ λ = I Tμ λ. 6 It s clear. For every λ, η [μ], λ η = I Tμ λ η I T μ λ Tμ η = I T μ λ I Tμ λ = λ η. 7 Let A be a subset of X and λ [μ]. The restrcton of λ on A s denoted by λ A. Theorem Let X, μ, be a μ-sts and A X. Defne a mappng A :[μ] L by A λ = η :η [μ], η A = λ. Then A s a μ-sts on A. It s clear that A χ = A μ = 1. λ 1, λ 2 L A, λ 1, λ 2 μ. A λ 1 A λ 2 = Tμ η1 : η1 [ μ ], η 1 A = λ 1 = Tμ η2 : η2 [ μ ], η 2 A = λ 2 η1 Tμ η2 : η1, η 2 [ μ ], η 1 η 2 A = λ 1 λ 2 Tμ η1 η 2 : η1, η 2 [ μ ], η 1 η 2 A = λ 1 λ 2 = A λ 1 λ 2. 9 I, A λ : I = η :η [μ], η A = λ. So A λ : I = Tμ η : I : η [ μ ], η A = λ Tμ η : I : η [ μ ], η A = λ = A λ : I. 10 Defnton The μ-sts A, μ, A scalledasubspaceof X, μ, and A s called the nduced μ-sts on A from. Theorem Let A, μ, A be a μ-smooth subspace of X, μ, and λ L A, λ [μ]. Then a F λ = F Tμ A η:η [μ], η A = λ, b f B A X, then B = A μ B.

3 Advances n Fuzzy Systems 3 a we have F λ = T Tμ A μ A λ [ ] = Tμ η : η μ, η A = λ = Tμ η : η [ μ ], η A = λ = FTμ η : η [ μ ], η A = λ = FTμ ξ : ξ [ μ ], ξ A = λ. b we have [ ] B λ = Tμ η : η μ, η B = λ [ ] = Tμ η : η μ, η A = ξ : ξ [ μ ] A, ξ B = λ = Tμ A ξ : ξ [ μ ] A, ξ B = λ = Tμ A. μ B Theorem 2.4. Let μ L X, γ L Y where L and L are copes of [0, 1] and f : X Y a h, μ, γ-fuzzy contnuous functons where f 1 γ = μ. Then for every γ-closed fuzzy set η, f 1 η s a μ-closed fuzzy set. Let η be γ-closed set. Then η s a γ-opensetandwe have Hence f 1 η = μ nf h 1 η f, μ = sup μ h 1 η f,0 f 1 η = f 1 γ η = μ h 1 η f = μ f 1 η. = h 1 γ η f μ = h 1 γ f h 1 η f μ = f 1 γ f 1 η = μ f 1 η = F Tμ f 1 η Relatve Smooth Contnuous Maps The concept of contnuty has been studed by Chang, Ramadan [4, 5] but here we shall study ths concept from a dfferent pont of vew. Defnton 2.1. Let h : L L be a lnear somorphsm of vector lattces or an order preservng one-toone mappng when L and L are copes of [0, 1] and X, μ,, Y, γ, T γ μ-sts and γ-sts, respectvely. A functon f : X Y s called h, μ, γ-smooth fuzzy contnuous f f 1 η T γ η forallη T γ,where f 1 ηx = h η f x μx forallx X. f 1 γ μ s called the nverse mage of γ relatve to μ. Remark 2.2. When L = L = [0, 1] and μ = χ X then the X, χ X, τ χx, χ X -RST concdes wth the fuzzy topologcal space X, τ defned by Chang [5 7]. Theorem 2.3. Let L = L = [0, 1] and f : X Y be I, χ X, χ Y -fuzzy contnuous, where I : L L s the dentty functon. Then f s contnuous n Chang s vew. In Remark 2.2 we consdered X, χ X, T χx and Y, χ Y, T χy as fuzzy topologcal spaces. Now let γ be an open set of smooth topology T χy. Then T χx f 1 γ = T χx I 1 γ f χ X 13 = T χx γ f TχX γ. So f s a fuzzy contnuous functon. So f 1 ηsaμ-closed fuzzy set. Theorem 2.5. Let X, μ, be relatve smooth topologcal spaces for = 1, 2, 3. If f : X 1 X 2 and g : X 2 X 3 are relatve smooth contnuous maps and μ 1 = f 1 μ 2 then so s g f. Usng the relatve smooth contnuty of g and f t follows that 1 g f 1 η = 1 f 1 g 1 η Snce for every x X, g f 1 η x = η g f x μ 1 x = η g f x μ 1 x μ 2 f x 2 g 1 η 3 η. 16 = g 1 η f x μ 1 x = f 1 g 1 η x = f 1 g 1 η x. 17 Theorem 2.6. Let X, μ, and Y, ν, T ν be two relatve smooth topologcal spaces, f : X Y arelatvesmooth contnuous map, A X, and f 1 ν = μ. Then the f A : A, μ A, A Y, ν, T ν s also relatve smooth contnuous.

4 4 Advances n Fuzzy Systems For each η [ν], 1 T μ A f A η = Tμ λ : λ [ μ ], λ A = 1 f A η f 1 η T ν η. 18 c If every λ T μ then λ T μ. Snce : λ Tμ : λ Tμ, I, 25 then τ μ I λ I τ μ λ The Representaton of a Relatve Smooth Topology Now we study the representaton of a relatve smooth topology. Let X, μ, beaμ-sts, L.Thenwedefne T μ = λ [ μ ] : Tμ λ. 19 Theorem 3.1. Let X, μ, be a μ-sts. Then for every > 0, Tμ s a relatve topologcal space. Moreover 1 2 mples T 2 T 1 μ μ. It s clear that χ, μ T μ. When λ, η T μ,wehave For τ μ beng a relatve L -fuzzy set, wth L = [0, 1], we can state a representaton theorem. Theorem 3.3. Let be a relatve smooth topology and Tμ the cut of. From the famles of relatve fuzzy topologes Tμ one bult T 1μ λ = : λ Tμ. Then T 1μ =. The proof s trval from the precedng results and the well-known fact that : λ T μ = : Tμ λ = λ. 27 Defnton 3.4. Let τ be a Chang fuzzy topology on X. Then a μ-smooth topology on X s sad to be compatble wth τ f τ =λ L X : λ μ > 0. and so λ, η, 20 Example 3.5. Let X be a nonempty set and :[μ] L be a mappng defned by μ = χ = 1, λ = 0for every λ [μ] \χ, μ. λ η Tμ λ η. 21 Ths mples that λ η Tμ. When λ j Tμ for each j I we have λ j λj. 22 Hence λ j Tμ. So Tμ s a relatve topology. Thesecondpartstrvaltoverfy,snceforλ Tμ 2, 2 1, λ Tμ 1,soTμ 1 Tμ 2. Theorem 3.2. Let Tμ, 0, 1] be a famly of μ-fuzzy topology on X such that 1 2 mples Tμ 1 Tμ 2.Letτ be the L -fuzzy set bult by τ μ λ = : λ Tμ. Then τ μ s a μ-smooth topology. a τ μ χ = τ μ μ = 1 by the defnton. b For every λ, η [μ] and>0fλ, η Tμ then λ η Tμ. Therefore : λ η T μ : λ T μ, η Tμ mples that 23 τ μ λ η τμ λ τ μ η. 24 It s clear that s the only relatve smooth topology on X compatble wth the ndscrete fuzzy topology of Chang. Example 3.6. Let X be a nonempty set and defne a mappng :[μ] L by μ = χ = 1, λ = for every λ [μ] \χ, μ. It s clear that s a μ-smooth topology on X compatble wth the dscrete fuzzy topology of Chang. 4. Relatve Smooth Compactness Defnton 4.1. Let X, μ, beaμ-sts. λ [μ], A, B [μ]. A s called a relatve cover of λ, f A λ partcularly, A s called a cover X, μ, fa s a cover of μ. A s called a μ-open cover of λ,fa s a famly of μ-open and A s a cover of λ. For a cover A of λ, B s called a subcover of λ,fb A and B s stll a cover of λ. Defnton 4.2. Let X, μ, beaμ-sts. For every [0, 1, a famly A [μ] s called an -cover, f for every λ A, λ ; A s called a μ-open -cover f A s a famly of μ-open set and A s a -cover; A 0 [μ] scalledasub-cover of A f A 0 A and A 0 s an -cover. Defnton 4.3. Let [0, 1. A μ-sts X, μ, s called -compact f every μ-open -cover has a fnte sub--cover.

5 Advances n Fuzzy Systems 5 Theorem 4.4. Let f : X, μ, Y, ν, T ν be an onto μ-smooth contnuous mappng and f 1 ν = μ. IfX, μ, s -compact then so s Y, ν, T ν. Let λ : I be a ν-open -cover of ν. Now consder the famly f 1 λ : I, snce f s μ-smooth contnuous, we have λ T ν T ν f 1 λ 28 f 1 λ T μ. It follows that f 1 λ : I s a μ-open -cover of μ. Snce X, μ, s-compact there exsts a fnte subset I 0 of I such that f 1 λ : I 0 s a μ-open -cover of X, μ,. Snce f s onto, then λ : I 0 s a ν-open -cover of Y, ν, T ν, whch concludes the proof. References [1] L. A. Zadeh, Fuzzysets, Informaton and Control, vol. 8, no. 3, pp , [2] M. H. Anvar and M. R. Molae, Genomc from the vewpont of observatonal modelng, WSEAS Transactons on Bology and Bomedcne, vol. 2, no. 2, pp , [3] M. R. Molae, Relatve sem-dynamcal systems, Internatonal Uncertanty, Fuzzness and Knowlege-Based Systems, vol. 12, no. 2, pp , [4]A.A.Ramadan, Smoothtopologcalspaces, Fuzzy Sets and Systems, vol. 48, no. 3, pp , [5] C. L. Chang, Fuzzy topologcal spaces, Mathematcal Analyss and Applcatons, vol. 24, no. 1, pp , [6] R. Lowenl, Fuzzy topologcal spaces and fuzzy compactness, Mathematcal Analyss and Applcatons, vol. 56, no. 3, pp , [7] L. Yng-Mng and L. Mao-Kang, Fuzzy Topology, World Scentfc, Sngapore, 1997.

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