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Int. J. Pure Appl. Sci. Technol., 16(1) (2013), pp. 61-68 International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in Research Paper On -Induced L-Fuzzy Topological Spaces Baby Bhattacharya1,* and Jayasree Chakraborty1 1 Department of Mathematics, NIT Agartala, India *Corresponding author, e-mail: ([email protected]) (Received: 5-3-13; Accepted: 8-4-13) Abstract: The aim of this paper is to study the concept of -induced L-fuzzy topological spaces. Some properties of the functions (namely Scott- -continuous function) are studied. The Scott- -continuous functions turn out to be the natural tool for studying the -induced L-fuzzy topological space. We also discuss the connections between some separations, countability and covering properties of an ordinary topological space and its corresponding -induced L-fuzzy topological space. Keywords: Fuzzy lattice, prime element, regular -subset, Scott- -continuous function, -induced L-fuzzy topological space. 1. Introduction The concept of induced fuzzy topological space was first introduced by Weiss in 1975 [16]. In 1976, Lowen [10] called these spaces as topologically generated spaces. In 1999, Aygun, Warner, Kudri [2] introduced the concept of completely induced L-fuzzy topological space. The concept of -subset was first introduced by E.P. Lane in [9] and the concept of regular -subset was introduced by Mack in 1970 [11]. The complement of a regular -subset is called a regular -subset. We introduced a new class of functions from a topological space ( X, T ) to a fuzzy lattice L with its Scott topology called Scott- -continuous functions [3]. In this paper we study some of their properties and characterizations. We also prove that the set (T) of Scott- -continuous functions from a topological space ( X, T ) to L with its Scott topology, is an L-fuzzy topology. Scott- -continuous functions turn out to be the natural tool for studying -induced L-fuzzy topological space. Then we discuss the connection between several properties of an ordinary topological space ( X, T ) and its corresponding -induced L-fuzzy topological space( X, (T )). Throughout this work X and Y will be non empty ordinary sets and L = L ( ˅, ˄,′) will denote a fuzzy lattice i.e. a complete completely distributive Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 62 lattice with a smallest element 0 and a largest element 1 (0 ≠ 1) and with an order-reversing involution a→a′ (a∈L). L is therefore a continuous lattice. Also LX will denote the lattice of L-fuzzy subsets of X. We will denote by 1A the characteristic function of the ordinary subset A of X. We State Some Definitions and Results for Ready References: Definition 1.1.[11]: A subset H of a topological space X is called a regular -subset if H is an intersection of a sequence of closed sets whose interiors contain H. Equivalently, if H = Gi = clXGi for i∈N ,where each Gi is open in X, then H is regular -subset of X . The complement of a regular -subset is called a regular -subset. Definition 1.2.[4]: Let ( X,T ) be a topological space and ∈X. A function :( X,T )→I is called a Scott continuous (or lower semi continuous) at ∈X iff for every α∈[0,1] with α <f ( ) there is a neighbourhood U of ʽ ʼ such that α <f (x) for every x∈U. Then f is called Scott continuous (or lower semi continuous) on X iff f is Scott continuous (or lower semi continuous) at every point of X. Definition 1.3.[14]: The set (T) of Scott-continuous functions from a topological space ( X,T ) to L with its Scott-topology is an L- fuzzy topology, called the induced L-fuzzy topology(IL-FT). Definition 1.4.[5]: An element p of L is called prime iff p ≠ 1 and whenever a,b∈L with a b≤ p then a≤ p or b≤ p. The set of all prime elements of L will be denoted by pr (L). In [15] Warner has determined the prime elements of the fuzzy lattice LX. Here pr(LX) = {xp: x∈X and p∈pr(L)} Where for each x∈X and each p∈pr(L) , xp: X→L is the fuzzy set defined by xp(y) = These xp are called the L-fuzzy points of X and we have xp is a member of an L-fuzzy set g and we write xp∈g iff g(x) ≰ p. Definition 1.5[5]: Let L be a complete lattice and x,y∈L. We say that x is way below y, in symbols x<<y, if and only if for every directed subset D of L with y≤∨D, there exists a d∈D such that x≤d. Proposition 1.6[15]: The set of the form {l∈L : δ<<l} are Scott-open. Result 1.7[12]: The set of the form { ∈L : p} where p∈pr(L) generates the Scott topology of L. Proposition 1.8[7]: An L-fts (X,τ) is said to be fuzzy regular iff for every p∈pr(L), for each x∈X and each closed L-fuzzy set f such that there is y∈X with yp∉f′ and f(x)=0,there exist open L-fuzzy sets g,h such that xp∈g for every yp∉f′, yp∈h and (∀z∈X) g(z)=0 or h(z)=0. Definition 1.9[1]: A topological space ( X,T ) is said to be almost regular iff for each non empty regular closed subset F of X and for each point x∈F′ , there exist disjoint open sets U and V such that x∈U and F⊂V. Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 63 Definition 1.10[2]: A topological space ( X,T ) is Hausdroff iff for every x, y∈X (x≠y),there exists regular open sets U and V such that x∈U, y∈V and U∩V=φ. Definition 1.11[13]: A topological space ( X,T ) is completely Hausdroff iff every distinct points x, y of X, there are open sets U and V such that x∈U, y∈V and Cl(U)∩Cl(V) = φ. Definition 1.12[15]: Let (X,τ) be an L-fuzzy topological space. A non empty sub family β of τ is a basis for τ iff for every xp∈pr(LX) and every f∈τ with xp∈f , there is a g∈β such that xp∈g≤f. Definition 1.13[8]: An L-fts (X,τ) is first countable C1,iff for every fuzzy point xp∈pr(LX), there exist a family of open L-fuzzy sets (fi)i∈N with xp∈ fi ,such that for every g∈τ with xp∈g, there is an i∈N with fi ≤g. 2. Scott- -Continuous Functions & their Properties Definition 2.1.[3]: Let ( X,T ) be a topological space and ∈X. A function :( X, T )→L where L has its Scott topology is said to be Scott- -continuous at ∈X iff for every p∈pr(L) with ( ) p , there is a regular -neighbourhood N of ʽ ʼ in ( X,T ) such that (x) p , x∈N . i.e. ({ ∈L: p}). Then is called Scott- -continuous on X iff is Scott- -continuous at every N⊂ point of X. Proposition 2.2: The characteristic function of every regular Proof: Let A be a regular A is defined as -subset is Scott- -continuous. -subset in a topological space ( X,T ) and the characteristic function 1A of 1A(x) = We have to show that 1A is Scott- -continuous i.e. 1A(x) p , for every x∈A and p∈pr(L). Let ∈X, p∈pr(L) with1A( ) p . Then A and A is a regular -neighbourhood of . We also have 1A(x) p, for all x∈A. Hence 1A is Scott- -continuous at ∈X. Proposition 2.3: If { j: j∈ } is an arbitrary family of Scott- -continuous functions from a topological space ( X, T ) L , then =˅j∈Λ j is also Scott- -continuous. Proof: Let p∈pr (L) and ∈X with ( ) =˅j∈Λ j( ) p , then there is a j∈Λ such that j ( ) p , since j is Scott- -continuous at ʽ ʼ , there is regular -neighbourhood N of ʽ ʼ such that j(x) p , x∈N. Hence (x) =˅j∈Λ j(x) p, x∈N. Thus is Scott- -continuous at ∈X. Proposition 2.4: Let (X,T) be a topological space. If , : ( X,T ) functions then : ( X,T ) L is Scott- -continuous as well. L are Scott- -continuous Proof: Let ∈X and p∈pr (L) with )( ) p. Then ( ) p and ( ) p. Since and are Scott- -continuous at ʽ ʼ, there are regular -neighbourhoods U and V of ʽ ʼ such that (x) p for all x∈U and (x) p for all x∈V. Let W = U V. Then W is a regular -neighbourhood of ʽ ʼ, Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 since the intersection of two regular ( (x) p for all x∈W. Hence 64 -subsets is regular -subset and p is prime. We have is Scott- -continuous at ∈X . Proposition 2.5: Let ( X,T ) be a topological space. The function :( X,T ) L is Scott- -continuous iff for every p∈pr (L), -1({ ∈L: p}) can be expressed as a union of some regular -subsets in ( X,T ). -1 Proof: Necessity: Let p∈pr(L) and x ({ ∈L: p}). Then (x) p. Since is Scott- continuous at x, then there exist a regular -subset Nx in (X,T) such that x∈Nx and -1 -1 Nx⊂ { ∈L: p}.Hence { ∈L: p}= Nx where Nx is regular -subset. Sufficiency: Let ∈X and p∈pr(L) with ( ) p .Then ∈ -1({ ∈L: is a regular -subset N in ( X,T ) such that ∈N and N⊂ -1({ ∈L: Scott- -continuous. Theorem 2.6: For a topological space ( X,T ) the set continuous} is an L-fuzzy topology on X. p}). By the hypothesis, there p}) , which implies that (T) ={ ∈LX: :( X,T ) is L is Scott- - Proof: From the proposition 2.2, 2.3 and 2.4, the proof is obvious. Remark: Since every Scott- -continuous function from a topological space ( X,T ) to a fuzzy lattice L is Scott continuous, we have (T) ⊂ (T), where (T) is the L-fuzzy topology of Scott continuous functions from (X,T) to L. Definition 2.7: The L-fuzzy topology defined above is called a -induced L-fuzzy topology ( -ILFT) and the space (X, ( T )) is called -induced L-fuzzy topological space and the members of (X, ( T )) are called fuzzy open sets in(X, ( T )) . Proposition 2.8: For a topological space ( X,T ), the following family forms a base for (β) = {Zβ : Z is regular where Zβ : X (T) -subset in ( X,T ), β ∈L} L is defined by Zβ(x)=β if x∈Z and Zβ(x)=0 otherwise. Proof: Let f∈ (T) and xp∈pr (LX) with (x) p.Then is Scott- -continuous and (x) p .By the continuity of L[5] there exists η∈L and η<<f(x) and η p. We take β∈L with η <<β <<f(x). Hence (x)∈{ ∈L: β <<l}. Since { ∈L: β <<l }is Scott open [by proposition1.6], there is a q∈pr(L) such that (x)∈{ ∈L: l q}⊂{ ∈L: β <<l} by proposition 1.7. Then (x) q. From the Scott- continuity of , there exist a regular -subset Z in (X,T) such that x∈ ∈Z and f(z) q,∀z∈Z. Thus β<<f(z) for every z∈Z and hence β ≤ f(z) for every z∈Z. Moreover, β p because β>>η p. β β β Consequently xp∈Z . Z ∈ (β) and Z ≤ f. Thus by proposition1.12 (β) is a base for (T). Lemma 2.9: If A is a regular -subset in a topological space ( X,T ) then 1A Proof: By proposition 2.2, the lemma follows immediately. (T) Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 65 Definition 2.10: A function f :( X,T1 )→( Y,T2 ) is called -irresolute if the inverse image f -1(A) is regular -subset in (X, T1) for every regular -subset in Y. Definition 2.11[3]: A function :(X, (T1)) space is said to be fuzzy continuous if -1( ) Theorem 2.12: A function :(X, -irresolute. Proof: A function :(X, Then 1A regular (T1)) (T1)) (Y, (T2). By fuzzy continuity (Y, (T2)) from a -IL-FT space to another (T1) for every (T2). (Y, (T2)) is fuzzy continuous iff (T2)) is fuzzy continuous and A is regular -1 (1A) =1 f −1 ( A) -subset in (X,T1). Let p∈pr(L) and x∈ -1 -IL-FT :(X,T1) (Y,T2) is -subset in (Y,T2). -1 (T1). Now we shall show that (A). Then 1 f −1 ( A) (x) p . Since 1 f −1 ( A) -subset N in (X,T1) such that x∈N and N⊂1 f −1 ( A) ({ ∈L: (T1), p}) = -1 -1 we have for each x (A) there exist a regular -subset N in (X,T1) such that x∈N⊂ -1 shows that (A) is regular -subset in (X,T1). Hence :(X,T1) (Y,T2) is -irresolute. -1 there exist a regular (A) is (A). Thus (A). This Conversely: Suppose :(X,T1) (Y,T2) is -irresolute. Let λ∈ (T2). We shall show that -1 ( ) (T1). i.e. -1( ):( X,T ) L is Scott- -continuous. Let ∈X and p∈pr(L) with -1 ( )( ) p. Then ( (a)) p. Since :(Y, T2) L is Scott- -continuous at ( )∈Y, there exist a regular -subset N in (Y,T2) such that ( )∈N and (y) p for all y∈N. Since N is regular -1 -subset in (Y,T2), -1(N) is regular -subset in (X,T1) by the hypothesis. Now we have (N), -1 -1 which implies there is a regular -subset B in (X,T1) such that B⊂ (N). Hence ( )(x) -1 = ( (x)) p for every x∈B. This shows that ( ) is Scott- -continuous. Consequently :(X, (T1)) (Y, (T2)) is fuzzy continuous. 3. Connection between Properties of a Topological Space and Its Corresponding - Induced L-Fuzzy Topological Space: In this section, we study the connections between separation, countability and covering properties of an ordinary topological space ( X,T ) and its corresponding -induced L-fuzzy topological space (X, ( T )) . Definition 3.1: A topological space ( X, T ) is called points x,y of X there are regular -completely Hausdroff iff for any distinct -subsets A and B such that x∈A , y∈B and Cl(A) Cl(B) = φ. Definition 3.2: A -ILFT space (X, (T)) is said to be fuzzy completely Hausdroff iff for every distinct points x,y of X and every p,q∈ pr (L) there exists L-fuzzy sets f and g such that xp∈f ,yq∈g and ∀z∈X , Cl(f)(z) = 0 or Cl(g)(z) = 0. Theorem 3.3: The topological space ( X, T ) is (X, (T )) is fuzzy completely Hausdroff. -completely Hausdroff iff the -IL-FT space Proof: Let x, y∈X(x y) and p, q∈pr(L). By -complete Hausdroffness of ( X,T ), there exists two regular -subsets U and V in (X,T), such that x∈U and y∈V and Cl(U) Cl(V) = φ.Then 1U,1V (T) Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 because U and V are regular -subsets in ( X,T ). We also have 1U(x) p, 1V(y) Cl(1U)(z) = 1Cl(U)(z) = 0 or Cl(1V)(z) = 1Cl(V)(z) =0 because Cl(U) Cl(V) =φ. ( X, ( T )) is a fuzzy completely Hausdroff. 66 q and z∈X. Consequently Conversely, let x,y∈X(x y) and p,q∈pr(L). From the fuzzy complete Hausdrofness of (X, ( T )), there exists basic open L-fuzzy sets , which are defined by respectively (z) = if z∈U, (z)=0 otherwise and (z)= if z V, (z)=0 otherwise, where U and V are regular -subsets in ( X,T ) and ∈L such that xp∈ , yq and ( z∈X), Cl( )(z) =0 or Cl( )(z) =0. Hence we have x∈U and y∈V where U and V are regular -subsets in ( X,T ) and Cl(U) Cl(V) =φ. Hence ( X,T ) is completely Hausdroff. Since the intersection of two regular -subsets is regular -subset, then the family of all regular -subsets in ( X,T ) forms a base for a smaller topology Tσ on X, called the -semi-regularization of T. A topological space( X,T ) is sad to be -semi-regular iff T = Tσ .i.e. (X, Tσ) space is the -semiregularization topological space. Obviously Tσ⊂T. Analogous to the almost regular [1] space we define the following: Definition 3.4: A topological space ( X,T ) is said to be -almost regular iff for each non empty regular -subset F of X and each point x∈F′, there exists disjoint open sets U and V such that x∈U and F⊂V. Obviously, every -almost regular topological space is almost regular. Proposition 3.5: A topological space ( X,T ) is -almost regular if for each non empty regular subset F of X and each point x∈F′, there exists disjoint regular -subset U and V such that x∈U and F⊂V. Proof: It is obvious from the definition of -almost regular. Since regular -subset implies open set. Lemma 3.6: Let ( X,T ) be a topological space and (X,Tσ) be its -semi-regularization topological space. Then (X,T ) is -almost regular iff (X, Tσ) is regular. Theorem 3.7: A topological space ( X,T ) is -almost regular iff the -induced L-fts (X, fuzzy regular. ( T )) is Proof: Let p∈pr (L), x∈X and let f be a closed L-fuzzy set in (X, ( T )) such that there is a y∈X with yp∉ f ′ and f(x) = 0. Then xp f ′ and f′∈ (T). So by the proposition2.8 there is a basic open L-fuzzy set g∈LX defined by g(z) = γ if z∈A and g(z) = 0 otherwise, where A is regular -subset in (X,T ) and γ∈L such that xp∈g≤ f ′ . Then, we have g(x) p which implies that g(x)= γ p and hence x∈A. Since A′ is regular -subset in ( X,T ) and ( X,T ) is -almost regular, then there are regular -subsets U,V in ( X,T ) such that x∈U, A′⊂V and U∩V=φ. Let η=1U and θ=1V . Then η,θ∈ (T), η(x) = 1 p and for every yp∉ f . yp∈θ because yp∈ f ′ implies that g(y)≤p and hence g(y)=0 because γ p. This means that y∈A′.Thus y∈V and θ(y)=1 p. In addition, (∀z∈X) η(z)=0 or θ(z)=0. Consequently (X, ( T )) is fuzzy regular. Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 67 Conversely, let K be a non-empty regular -subset in ( X,T ) and let x∈K. Then 1K is closed in (X, ( T )) . 1K(x) = 0 and there exist a y∈X with yp∉ 1′K because K is non-empty set. From the fuzzy regularity of (X, ( T )) , there exist basic open L-fuzzy sets g,h∈LX defined by g(z) = γ if z∈A and -subsets in g(z) = 0 otherwise and h(z)=β if z∈B and h(z) = 0 otherwise, where A and B are regular ( X,T ) and γ,β∈L such that xp∈g, yp∈h for every yp∉ 1′K and (∀z∈X) g(z) = 0 or h(z) = 0. Hence x∈A, K⊂ B and A∩B = φ. Thus by the proposition 3.5 ( X,T ) is -almost regular. Definition 3.8: A topological space ( X,T ) is called -nearly compact iff every regular has a finite subcover. -cover of X Definition 3.9.[12]: An L-fts ( X,T ) is said to be fuzzy compact iff for every prime element p of L and every collection( fi)i∈J of open L–fuzzy sets with (˅i∈J fi)(x) p for all x∈X, there is a finite subset F of J such that (˅i∈F fi)(x) p) for all x∈X. Lemma 3.10: A topological space ( X,T ) is -nearly compact iff (X, definition 3.9[12] we prove the following theorem. Theorem 3.11: A -nearly compact. ) is compact .Using the -ILFT space is fuzzy compact iff the corresponding topological space ( X,T ) is Proof: Let us assume that ( X,T ) is -nearly compact. Let p∈pr (L) and = { fj : j } be a family of basic open L- fuzzy sets in (X, ( T )) with ( fj (x) p for all x∈X; where fj(x) = if x∈Aj and fj(x)= 0 otherwise, then Aj are regular each x∈X there is j∈ such that fj(x) -subset in ( X,T ) and p i.e. p and let L for every j = { Aj : there is an j . Then for such that p and fj }. Then is a family of regular -subsets in ( X,T ) covering X. In other words, is a regular -cover of X. From the -compactness of ( X,T ), there exists a finite subfamily of say where = {A1,A2,…,.An} such that X = fj) (x) p for all x∈X and thus n. Hence ( (X, ( T )) is fuzzy compact. Conversely, suppose (X, 1A (T) for every j L-fuzzy sets in (X, (X, (T )) is fuzzy compact. Let {Aj: j and Aj is regular ( T )) with Hence X = -subset in ( X,T ). Thus {1 A j : j 1 A j )(x) ( T )) there exist a finite subset } be a regular of -cover of X. Then } is a family of open p for all x∈X. From the fuzzy compactness of such that 1 A j )(x) p for all x∈X . Aj. Thus ( X,T ) is -nearly compact. Theorem 3.12: Let ( X,T ) be a topological space. Then the -semi regularization topological space (X,Tσ) is first countable (C1) iff the 1.13(8)]. -induced L-fts ( X, ( T )) is first countable.[Definition Proof: Assume that (X,Tσ) is first countable. Let xp be any fuzzy point and Γ be a countable local base of x in (X,Tσ ). Let = { Aγ : A∈ Γ ,γ∈L with γ p}, where Aγ (z) = γ if z∈A and Aγ (z) = 0 otherwise. We shall show that is a countable local base of xp in (X, ( T )) . Let g∈ (T) and xp∈g. Then, by proposition 2.8, there exists a basic open L-fuzzy set Zβ such that xp∈Zβ ≤ g, where Z is regular β subset in ( X,T ). Thus, Z (x) = β p and hence x∈Z. Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68 68 Since Z Tσ , x∈Z and Γ is a local base of x in (X,Tσ), there exists an A∈ Γ such that x∈A⊂Z. Then, we have Aβ and xp ≤ Aβ≤ Zβ ≤ g and therefore is a local base of xp. Moreover, is a countable local base because is countable. Consequently, (X, (T)) is first countable. be the countable local Now assume that (X, ( T )) is first countable. Let x∈X and p∈pr(L) and -1 base of xp in (X, (T)) . Consider Γ = { ({ ∈L: p}): f ∈ Γ }. We will prove that Γ is countable local base of x in (X,Tσ). Let O∈ Tσ and x∈O. Then, there is a regular -subset A in (X,T ) such that x∈A⊂O. Hence, 1A (T) and 1A (x) = 1 p since is a local base of xp, there is f∈ such that -1 xp∈f ≤ 1A and so x ({ ∈L: p})⊂A⊂O. Thus Γ is a local base of x in (X,Tσ). Since is a countable, Γ is countable and therefore (X,Tσ) is first countable. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] M.K. Singhal and S.P. 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