WEAKLY PREOPEN AND WEAKLY PRECLOSED FUNCTIONS IN FUZZY TOPOLOGY*

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1 International Journal of General Toology Vol. 4, Nos. 1-2, January-December 2011, ; ISSN: WEAKLY PREOPEN AND WEAKLY PRECLOSED FUNCTIONS IN FUZZY TOPOLOGY* Miguel Caldas 1, Govindaa Navalagi 2, and Ratnesh Saraf 3 ABSTRACT In this aer, we introduce and characterize fuzzy weakly reoen and fuzzy weakly reclosed functions between fuzzy toological saces and also study these functions in relation to some other tyes of already known functions. *2000 Math. Subject Classification: Primary: 54A40; Keywords and Phrases: Fuzzy reoen sets, fuzzy weakly oen, fuzzy weakly-continuous, fuzzy strongly continuous, f.a.o.n. functions, fuzzy re-connected saces. 1. INTRODUCTION AND PRELIMINARIES The concet of fuzzy sets was introduced by Prof. L.A. Zadeh in his classical aer [19]. After the discovery of the fuzzy subsets, much attention has been aid to generalize the basic concets of classical toology in fuzzy setting and thus a modern theory of fuzzy toology is develoed. The notion of fuzzy subsets naturally lays a very significant role in the study of fuzzy toology which was introduced by C.L. Chang [4] in In 1980, Ming and Ming [9], introduced the concets of quasi-coincidence and q-neighbourhoods by which the extensions of functions in fuzzy setting can very interestingly and effectively be carried out. In 1985, D.A. Rose [16] defined weakly oen functions in toological saces. In 1997 J.H. Park, Y.B. Park and J.S. Park [13] introduced the notion of weakly oen functions in between fuzzy toological saces. In this aer we introduce and discuss the concet of fuzzy weakly reoen functions which is weaker than fuzzy reoen and f.a.o.n functions introduced by [3] and Nanda [11] resectively and we obtained several characterizations and roerties of these functions. We also study these functions comaring with other tyes of already known functions. Here is seen that fuzzy reoenness imlies fuzzy weakly reoenness but not conversely. But under a certain condition the converse is also true. We also introduce and study the concet of fuzzy weakly reclosed functions. Throughout this aer by (X, τ) or simly by X we mean a fuzzy toological sace (fts, shorty) due to Chang [4]. A fuzzy oint in X with suort x X and value (0 < 1) is denoted by x. Two fuzzy sets λ and β are said to be quasi-coincident (q-coincident, shorty) denoted by λqβ, if there exists x X such that λ(x) + β(x) > 1 [9] and by q we denote is not q-coincident. It is known [9] that λ β if and only if λq (1 β). A fuzzy set λ is said to be q-neighbourhood (q-nbd) of x if there is a fuzzy oen set µ such that x qµ.

2 38 International Journal of General Toology The interior, closure, and the comlement of a fuzzy set λ X are denoted by Int(λ), Cl(λ) and 1 λ resectively. For definitions and results not exlained in this aer, the reader is referred to [1, 4, 8, 10, 14, 16, 19] assuming them to be well known. Definitions 1.1: A fuzzy set λ in a fts X is called, 1. Fuzzy reoen [3] if λ Int(Cl(λ)). 2. Fuzzy reclosed [3] if Cl(Int(λ)) λ. 3. Fuzzy regular oen [1] if λ = Int(Cl(λ)). 4. Fuzzy regular closed [1] if λ = Cl(Int(λ)). 5. Fuzzy α-oen [3] if Int(Cl(Int(λ))). 6. Fuzzy α-closed [3] if Cl(Int(Cl(λ))) λ. Recall that if, λ be a fuzzy set in a fts X then Cl(λ) = {β : β λ, β is fuzzy reclosed} (res. Int(λ) = {β : λ β, β is fuzzy reoen}) is called a fuzzy reclosure of λ (res. fuzzy reinterior of λ) [3]. Result 1.2: 1. A fuzzy set λ in a fts X is fuzzy reclosed (res. fuzzy reoen) if and only if λ = Cl(λ) (res. λ = Int(λ)) [3]. 2. Let λ a fuzzy set of a fts (X, τ). Then; (i) Int(λ) Int(λ) λ Cl(λ) Cl(λ). (ii) 1 X Int(λ) = Cl(1 X λ) and Int(1 X λ) = 1 Cl(λ). Definition 1.3: Let f : (X, τ 1 ) be a function from a fts (X, τ 1 ) into a fts (X, τ 2 ). The function f is called: (i) fuzzy reclosed [3] if f(λ) is a fuzzy reclosed subset of Y for each fuzzy closed set λ in X: (ii) fuzzy oen [4] if f(λ) is a fuzzy oen subset of Y for each fuzzy oen subset λ of X. (iii) fuzzy weakly oen [13] if f(λ) Int(f(Cl(λ))) for each fuzzy oen set λ in X. (iv) fuzzy almost oen (written as f.a.o.n) [11] if f(λ) is a fuzzy oen subset of Y for each fuzzy regular oen set λ in X. (v) fuzzy α-oen [8] if f(λ) is a fuzzy α-oen set of Y for each fuzzy oen subset λ of X. (vi) fuzzy θ-continuous [15] (res. fuzzy weakly θ-continuous [15]) if for each fuzzy oint x and each oen nbd λ of of f(x ), there is an fuzzy oen nbd µ of x such that f(cl(µ)) Cl(λ) (res. f(int(cl(µ))) Cl(λ)). (vii) fuzzy contra-oen (res. fuzzy contra closed) if f(λ) is a fuzzy closed (res. fuzzy oen) set of Y for each fuzzy oen (res. fuzzy closed) set λ in X. Definitions 1.4: [10]. A fuzzy oint x in a fts X is said to be a fuzzy θ-cluster oint of a fuzzy set λ if and only if for every fuzzy oen q-nbd µ of x, Cl(µ) is q-coincident with λ. The set of all fuzzy θ-cluster oints of λ is called the fuzzy θ-closure of λ and is denoted by Cl θ (λ). A fuzzy set λ is fuzzy θ-closed if and only if λ = Cl θ (λ). The comlement of a fuzzy θ-closed set is called of fuzzy θ-oen and the θ-interior of λ denoted by Int θ (λ) is defined as:

3 Weakly Preoen and Weakly Preclosed Functions in Fuzzy Toology* 39 Int θ (λ) = {x : for some fuzzy oen q-nbd, Β of x,cl(β) λ}. Lemma 1.5: [2]. Let λ be a fuzzy set in a fts X, then: 1. λ is a fuzzy θ-oen if and only if λ = Int θ (λ) Int θ (λ) = Cl θ (1 λ) and Int θ (1 λ) = 1 Cl θ (λ). 3. Cl θ (λ) (res. Int θ (λ)) is a fuzzy closed set (res. fuzzy oen set) but not necessarily is a fuzzy θ-closed set (res. fuzzy θ-oen set). Result. 1.6: (i) It is easy to see that Cl(λ) Cl θ (λ) and Int θ (λ) Int(λ) for any fuzzy set λ in a fts X. (ii) For a fuzzy oen (res. fuzzy closed) set λ in a fts X, Cl(λ) = Clθ(λ) (res. Int θ (λ) = Int(λ)). 2. FUZZY WEAKLY PREOPEN FUNCTIONS Now, we define the generalized form of reoen functions in fuzzy setting. Definition 2.1: A function f : (X, τ 1 ) is said to be fuzzy weakly reoen if f(λ) Int(f(Cl(λ))) for each fuzzy oen subset λ of X. Clearly, every fuzzy weakly oen function is fuzzy reoen and every fuzzy reoen function is fuzzy weakly reoen (since, f(λ) = Int(f(λ)) Int(f(Cl(λ))) for each fuzzy oen set λ of X), but the converse is not generally true, as the next examle shows. Examle 2.2: Let X = {a, b, c}, τ = {0, A, B, A B, A B, 1} and σ = {0, E, H, E H, E H, 1}. Where A, B, E and H are defined as follows: A(a) = 0.4, A(b) = 0.7, A(c) = 0.2; B(a) = 0.3, B(b) = 0.1, B(c) = 0.6; E(a) = 0.5, E(b) = 0.8, E(c) = 0.3; H(a) = 0.4, H(b) = 0.2, H(c) = 0.7. Consider the identity maing f : (X, τ) (X, σ). Clearly f is fuzzy weakly reoen but not fuzzy reoen. Examle 2.3: Let X = {a, b, c} and Y = {x, y, z}, τ = {0, A, 1}, σ = {0, B, H, 1}. Fuzzy sets A, B, and H be defined as : A(a) = 0.5, A(b) = 0.3, A(c) = 0.2; B(x) = 0.9, B(y) = 1, B(z) = 0.7; H(x) = 0.2, H(y) = 0.9, H(z) = 0.3. Then the maing f : (X, τ) (Y, σ) defined by f(a) = z and f(b) = x, f(c) = y is fuzzy weakly reoen but not fuzzy weakly oen. Theorem 2.4: For a function f : (X, τ 1 ), the following conditions are equivalent: (i) f is fuzzy weakly reoen,

4 40 International Journal of General Toology (ii) f(int θ (λ)) Int(f(λ)) for every fuzzy subset λ of X, (iii) Int θ (f 1 (β)) f 1 (Int(β)) for every fuzzy subset β of Y, (iv) f 1 (Cl(β)) Cl θ (f 1 (β)) for every fuzzy subset β of Y. Proof: (i) (ii) : Let λ be any fuzzy subset of X and x a fuzzy oint in Int θ (λ). Then, there exists a fuzzy oen q-nbd γ of x such that γ Cl(γ) λ. Then, f(γ) f(cl(γ)) f(λ). Since f is fuzzy weakly reoen, f(γ) Int(f(Cl(γ))) Int(f(λ)). It imlies that f(x ) is a oint in Int(f(λ)). This shows that x f 1 (Int(f(λ))). Thus Int θ (λ) f 1 (Int(f(λ))), and so f(int θ (λ)) Int(f(λ)). (ii) (i) : Let µ be a fuzzy oen set in X. As µ Int θ (Cl(µ)) imlies, f(µ) f(int θ (Cl(µ))) Int(f(Cl(µ))). Hence f is fuzzy weakly reoen. (ii) (iii) : Let β be any fuzzy subset of Y. Then by (ii), f(int θ (f 1 (β))) Int(β). Therefore Int θ (f 1 (β)) f 1 (Int(β)). (iii) (ii) : This is obvious. (iii) (iv) : Let β be any fuzzy subset of Y. Using (iii), we have 1 Cl θ (f 1 (β)) = Int θ (1 f 1 (β)) = Int θ (f 1 (1 β)) f 1 (Int(1 β)) = f 1 (1 Cl(β)) = 1 (f 1 (Cl(β)). Therefore, we obtain f 1 (Cl(β)) Cl θ (f 1 (β)). (iv) (iii): Similary we obtain, 1 f 1 (Int(β)) 1 Intθ(f 1 (β)), for every fuzzy subset β of Y, i.e., Int θ (f 1 (β)) f 1 (Int(β)). Theorem 2.5: If X is a fuzzy regular sace, then for a function f : (X, τ 1 ), the following conditions are equivalent: (i) f is fuzzy weakly reoen, (ii) For each fuzzy θ-oen set λ in X, f(λ) is fuzzy reoen in Y, (iii) For any fuzzy set β of Y and any fuzzy θ-closed set λ in X containing f 1 (β), there exists a fuzzy reclosed set d in Y containing β such that f 1 (δ) λ. Proof: (i) (ii) : Let λ be a fuzzy θ-oen set in X. Then 1 f(λ) is a fuzzy set in Y and by (i) and Theorem 2.4 (iv), f 1 (Cl(1 f(λ))) Cl θ (f 1 (1 f(λ))). Therefore, 1 f 1 (Int(f(λ))) Cl θ (1 λ) = 1 λ. Then, we have λ f 1 (Int(f(λ))) which imlies f(λ) Int(f(λ)). Hence f(λ) is fuzzy reoen in Y. (ii) (iii) : Let β be any fuzzy set in Y and λ be a fuzzy θ-closed set in X such that f 1 (β) λ. Since 1 λ is fuzzy θ-oen in X, by (ii), f(1 λ) is fuzzy reoen in Y. Let δ = 1 f(1 λ). Then δ is fuzzy reclosed and β δ. Now, f 1 (δ) = f 1 (1 f(1 λ)) = 1 f 1 (f(1 λ)) λ. (iii) (i) : Let β be any fuzzy set in Y. Then by Corollary 3.6 of [10] λ = Cl θ (f 1 (β)) is fuzzy θ-closed set in X and f 1 (β) λ. Then there exists a fuzzy reclosed set δ in Y containing β such that f 1 (δ) λ. Since δ is fuzzy reclosed f 1 (Cl(β)) f 1 (δ) Cl θ (f 1 (β)). Therefore by Theorem 2.4, f is a fuzzy weakly reoen function. Theorem 2.6: For a function f : (X, τ 1 ), the following statement are equivalent: (i) f is fuzzy weakly reoen, (ii) For each x fuzzy oint in X and each fuzzy oen subset µ of X containing x, there exists a fuzzy reoen set δ containing f(x ) such that δ f(cl(µ)).

5 Weakly Preoen and Weakly Preclosed Functions in Fuzzy Toology* 41 Proof: (i) (ii) : Let x X and µ be a fuzzy oen set in X containing x. Since f is fuzzy weakly reoen f(µ) Int(f(Cl(µ))). Let δ = Int(f(Cl(µ))): Hence δ f(cl(µ)), with δ containing f(x ). (ii) (i) : Let µ be a fuzzy oen set in X and let y f(µ). It following from (ii) that δ f(cl(µ)) for some δ fuzzy reoen in Y containing y. Hence we have, y δ Int(f(Cl(µ))). This shows that f(µ) Int(f(Cl(µ))), i.e., f is a fuzzy weakly reoen function. Theorem 2.7: Let f : (X, τ 1 ) be a bijective function. Then the following statements are equivalent. (i) f is fuzzy weakly reoen, (ii) Cl(f(λ)) f(cl(λ)) for each λ fuzzy oen subset of X, (iii) Cl(f(Int(β)) f(β) for each β fuzzy closed subset of X. Proof: (i) (iii) : Let β be a fuzzy closed set in X. Then we have f(1 β) = 1 f(β) Int(f(Cl(1 β))) and so 1 f(β) 1 Cl(f(Int(β))). Hence Cl(f(Int(β))) f(β). (iii) (ii) : Let λ be a fuzzy oen set in X. Since Cl(λ) is a fuzzy closed set and λ Int(Cl(λ)) by (iii) we have Cl(f(λ)) Cl(f(Int(Cl(λ))) f(cl(λ)). (ii) (iii) : Obvious. (iii) (i) : Similar to (iii) (ii). The roof of the following theorem is mostly straightforward and hence is omitted. Theorem 2.8: For a function f : (X, τ 1 ) the following conditions are equivalent: (i) f is fuzzy weakly reoen. (ii) f(int(β)) Int(f(β)), for each fuzzy closed subset β of X. (iii) f(int(cl(λ))) Int(f(Cl(λ))), for each fuzzy oen subset λ of X. (iv) f(λ) Int(f(Cl(λ))), for every fuzzy reoen subset λ of X. (v) f(λ) Int(f(Cl(λ))), for every fuzzy α-oen subset λ of X. Now, we give a definition of strong continuity. This definition when combined with fuzzy weak reoenness imlies fuzzy reoenness. Definition 2.9: A function f : (X, τ 1 ) is said to be fuzzy strongly continuous, if for every fuzzy subset λ of X, f(cl(λ)) f(λ). Lemma 2.10: If f : (X, τ 1 ) is fuzzy strongly continuous, then Int(f(Cl(λ))) f(λ), but the converse does not hold. Examle 2.3 above serves the urose. Theorem 2.11: If f : ( X, τ 1 ) ( Y, τ 2 ) is fuzzy weakly reoen and fuzzy strongly continuous, then f is fuzzy reoen. Proof: Let λ be a fuzzy oen subset of X. Since f is fuzzy weakly reoen f(λ) Int(f(Cl(λ))). However, because f is fuzzy strongly continuous, f(λ) Int(f(λ)) and therefore f(λ) is fuzzy reoen. A function f : (X, τ 1 ) is said to be fuzzy contra reclosed if f(λ) is a fuzzy reoen set of Y, for each fuzzy closed set λ in X.

6 42 International Journal of General Toology Theorem 2.12: If f : (X, τ 1 ) is fuzzy contra reclosed, then f is a fuzzy weakly reoen function. Proof: Let λ be a fuzzy oen subset of X. Then, we have f(λ) f(cl(λ))) = Int(f(Cl(λ))). The converse of Theorem 2.12 does not hold. Examle 2.13: Let X = {a, b, c} and Y = {x, y, z}. Define fuzzy sets A and B as : A(a) = 0, A(b) = 0.2, A(c) = 0.7; B(x) = 0, B(y) = 0.2, B(z) = 0.2. Let τ = {0, A, 1}, σ = {0, B, 1}. Then the maing f : (X, τ) (Y, σ) defined as : f(a) = x, f(b) = y and f(c) = y is fuzzy weakly reoen but not fuzzy contra reclosed. Theorem 2.14: Let X be a fuzzy regular sace. Then f : (X, τ 1 ) is fuzzy weakly reoen if and only if f is fuzzy reoen. Proof: The sufficiency is clear. For the necessity, let λ be a non-null fuzzy oen subset of X. For each x fuzzy oint in λ, let µ x be an fuzzy oen set such that x µ x Cl( µ x ) λ. Hence we obtain that λ = { µ x : x } {() : } λ = Cl µ x x λ and, f(λ) = {f( µ x ) : x λ} {Int(f(Cl( µ x ))) : x λ} Int(f( {Cl( µ x ) : x λ}) = Int(f(λ)). Thus f is fuzzy reoen. Theorem 2.15: If f : (X, τ 1 ) is an f.a.o.n function, then it is a fuzzy weakly reoen function. Proof: Let λ be a fuzzy oen set in X. Since f is f.a.o.n and Int(Cl(λ)) is fuzzy regular oen, f(int(cl(λ))) is fuzzy oen in Y and hence f(λ) f(int(cl(λ)) Int(f(Cl(λ))) Int(f(Cl(λ))). This shows that f is fuzzy weakly reoen. The converse of Theorem 2.15 is not true in general. Examle 2.2 above serves the urose. Lemma 2.16: [9] If f : (X, τ 1 ) is a fuzzy continuous function, then for any fuzzy subset λ of X, f(cl(λ)) Cl(f(λ)). Theorem 2.17: If f : (X, τ 1 ) is a fuzzy weakly reoen and fuzzy continuous function, then f is a fuzzy α-oen function. Proof: Let λ be a fuzzy oen set in X. Then by fuzzy weak reoenness of f, f(λ) Int(f(Cl(λ))). Since f is fuzzy continuous f(cl(λ)) Cl(f(λ)). Hence we obtain that, f(λ) Int(f(Cl(λ))) Int(Cl(f( ))) Int(Cl(Int(f(λ)))). Therefore, f(λ) Int(Cl(Int(f(λ))) which shows that f(λ) is a fuzzy α-oen set in Y. Thus by definition 1.3, f is a fuzzy α-oen function. Since, every fuzzy strongly continuous function is fuzzy continuous we have the following corollary, Corollary 2.18: If f : (X, τ 1 ) is a fuzzy weakly reoen and fuzzy strongly continuous function. Then f is a fuzzy α-oen function. Definition 2.19: Two non-emty fuzzy sets λ and β in a fuzzy toological saces X (i.e., neither λ nor β is 0 X ) are said to be fuzzy re-searated if λqcl() β and βqcl() λ or

7 Weakly Preoen and Weakly Preclosed Functions in Fuzzy Toology* 43 equivalently if there exist two fuzzy reoen sets µ and v such that λ µ, β v, and β qµ. A fuzzy toological sace X which cannot be exressed as the union of two fuzzy re-searated sets is said to be a fuzzy re-connected sace. Theorem 2.20: If f : (X, τ 1 ) is an injective fuzzy weakly reoen function of a sace X onto a fuzzy re-connected sace Y, then X is fuzzy connected. Proof: If ossible, let X be not connected. Then there exist fuzzy searated sets β and γ in X such that X = β γ. Since β and γ are fuzzy searated, there exist two fuzzy oen sets µ and v such that β µ, γ v, β q µ and γ q µ. Hence we have f(β) f(µ), f(γ) f(v), f(β) q f(v) and f(γ) q f(µ). Since f is fuzzy weakly reoen, we have f(µ) Int(f(Cl(µ))) and f(v) Int(f(Cl(v))) and since µ and v are fuzzy oen and also fuzzy closed, we have f(cl(µ)) = f(µ), f(cl(v)) = f(v). Hence f(µ) and f(v) are fuzzy reoen in Y. Therefore, f(β) and f(γ) are fuzzy re-searated sets in Y and Y = f(x) = f(β γ) = f(β) f(γ). Hence this contrary to the fact that Y is fuzzy re-connected. Thus X is fuzzy connected. Definition 2.21: A sace X is said to be fuzzy hyer-connected if every non-null fuzzy oen subset of X is fuzzy dense in X. Theorem 2.22: If X is a fuzzy hyer-connected sace, then a function f : (X, τ 1 ) is fuzzy weakly reoen if and only if f(x) is fuzzy reoen in Y. Proof: The sufficiency is clear. For the necessity observe that for any fuzzy oen subset λ of X, f(λ) f(x) = Int(f(X) = Int(f(Cl(λ))). 3. FUZZY WEAKLY PRECLOSED FUNCTIONS Now, we define the generalized form of reclosed functions in fuzzy setting. Definition 3.1: A function f : ( X, τ 1 ) ( Y, τ 2 ) is said to be fuzzy weakly reclosed if Cl(f(Int(β))) f(β) for each fuzzy closed subset β of X. Clearly, every fuzzy reclosed function is fuzzy weakly reclosed function since Cl(f(Int(β))) Cl(f(β)) = f(β) for every fuzzy closed subset β of X, but the converse is not generally true, as the next examle shows. Examle 3.2: Let X = {a, b} and Y = {x, y}. Fuzzy sets A and B are defined as : A(x) = 0.4, A(y) = 0.3; B(a) = 0.5, B(b) = 0.6. Let τ 1 = {0, B, 1 X } and τ 2 = {0, A, 1 Y }. Then the function f : (X, τ 1 ) defined by f(a) = x, f(b) = y is fuzzy weakly reclosed but not fuzzy reclosed. Recall that, every fuzzy closed function is fuzzy α-closed and every fuzzy α-closed function is fuzzy reclosed, but the reverse imlications not be true in general [5]. From result above and Examle 3.2, we have the following diagrama and the converses of these imlication do not hold, in general as is showd. λqv

8 44 International Journal of General Toology fuzzy closed function fuzzy α-closed function fuzzy weakly reclosed function fuzzy reclosed function Theorem 3.3: For a function f : (X, τ 1 ). The following conditions are equivalent. (i) f is fuzzy weakly reclosed, (ii) Cl(f(λ)) f(cl(λ)) for every fuzzy oen set λ in X. Proof: (i) (ii) : Let λ be any fuzzy oen subset of X. Then, Cl(f(λ)) = Cl(f(Int(λ))) Cl(f(Int(Cl(λ))) f(cl(λ)). (ii) (i) : Let β be any fuzzy closed subset of X. Then, Cl(f(Int(β))) f(cl(int(β))) f(cl(β)) = f(β). The roof of the following theorem, is mostly straightforward and hence is omitted. Theorem 3.4: For a function f : (X, τ 1 ) the following conditions are equivalent: (i) f is fuzzy weakly reclosed. (ii) Cl(f(β)) f(cl(β)) for each fuzzy oen subset β of X. (iii) Cl(f(Int(β))) f(β) for each fuzzy closed subset β of X. (iv) Cl(f(Int(β))) f(β) for each fuzzy reclosed subset β of X. (v) Cl(f(Int(β))) f(β) for every fuzzy α-closed subset β of X. Theorem 3.5: For a function f : (X, τ 1 ) the following conditions are equivalent: (i) f is fuzzy weakly reclosed. (ii) Cl(f(λ)) f(cl(λ)) for each fuzzy regular oen subset λ of X. (iii) For each fuzzy subset β of Y and each fuzzy oen set µ in X with f 1 (β) µ, there exists a fuzzy reoen set δ in Y with β δ and f 1 (δ) Cl(µ). (iv) For each fuzzy oint y in Y and each fuzzy oen set µ in X with f 1 (y ) µ, there exists a fuzzy reoen set δ in Y containing y and f 1 (δ) Cl(µ). (v) Cl(f(Int(Cl(λ)))) f(cl(λ)) for each fuzzy set λ in X. (vi) Cl(f(Int(Cl θ (λ)))) f(cl θ (λ)) for each fuzzy set λ in X. (vii) Cl(f(λ)) f(cl(λ)) for each fuzzy reoen set λ in X. Proof: It is clear that: (i) (vi), (iii) (iv), and (i) (v) (vii) (ii) (i): To show that (ii) (iii): Let β be a fuzzy subset of Y and let µ be fuzzy oen in X with f 1 (β) µ. Then f 1 (β) q Cl(1 X Cl(µ)) and consequently, β q f(cl(1 X Cl(µ))). Since 1 X Cl(µ) is fuzzy regular oen, β qcl(f(1 X Cl(µ))) by (ii). Let δ = 1 Y Cl(f(1 X Cl(µ))). Then δ is fuzzy reoen with β δ and f 1 (δ) 1 X f 1 (Cl(f(1 X Cl(µ)))) 1 X f 1 f(1 X Cl(µ)) Cl(µ). (vi) (i) : It is suffices see that Cl θ (λ) = Cl(λ) for every fuzzy oen sets λ in X. (iv) (i) : Let β be fuzzy closed in X and let y 1 Y f(β). Since f 1 (y ) 1 X β, there exists a fuzzy reoen δ in Y with y δ and f 1 (δ) Cl(1 X β) = 1 X Int(β) by (iv). Therefore δ q f(int(β)), so that y 1 Y Cl(f(Int(β))). Thus (iv) (i). Finally, for

9 Weakly Preoen and Weakly Preclosed Functions in Fuzzy Toology* 45 (vi) (vii): Note that Cl θ (λ) = Cl(λ) for each fuzzy reoen subset λ in X [6]. Theorem 3.6: If f : (X, τ 1 ) is fuzzy weakly reclosed, then for each fuzzy oint y in Y and each fuzzy oen q-nbd µ of f 1 (y ) in X, there exists a fuzzy reoen q-nbd δ of y in Y, such that f 1 (δ) Cl(µ). Proof: Let µ be any fuzzy oen q-nbd of f 1 (y ) in X. Then µ(x) + > 1 and hence there exists a ositive real number α such that µ(x) > α > 1. Then µ is a fuzzy oen q-nbd of f 1 (y α ). By Theorem 3.5 (iv) there exists a fuzzy reoen set δ containing y α in Y such that f 1 (δ) Cl(µ). Now, δ(y) > α and hence δ(y) > 1. Thus δ is a fuzzy reoen q-nbd of y. Definition 3.7: A fuzzy set λ in a fts (X, τ) is called re-q-nbd of x α if there exist a fuzzy reoen subset µ in X such that x α qµ λ. Theorem 3.8: In a fts (X, τ) a fuzzy oint x α Cl(λ) if and only if every re-q-nbd of x α is quasi-coincident with λ. Proof: Suose x α Cl(λ) and if ossible, let there exist a re-q-nbd µ of x α such that µ qλ. Then there exist a fuzzy reoen set v in X such that x α qv µ which shows that vqλ and hence λ 1 v. As 1 v is fuzzy reclosed, Cl(λ) 1 v. Since x α 1 v, we obtain x α Cl(Λ) which is a contradiction. Conversely, suose x α Cl(λ). Then there exists a fuzzy reclosed set λ β such that x α β. We then have x α q(1 β) FPO(X, τ) and λ q(1 β). Next we investigate conditions under which fuzzy weakly reclosed functions are fuzzy reclosed. Theorem 3.9: If f : (X, τ 1 ) is fuzzy weakly reclosed and if for each fuzzy closed subset β of X and each fiber f 1 (y ) 1 X β there exists a fuzzy oen q-nbd µ of X such that f 1 (y ) µ Cl(µ) 1 X β. Then f is fuzzy reclosed. Proof: Let β is any fuzzy closed subset of X and let y 1 Y f(β). Then f 1 (y ) q β and hence f 1 (y ) 1 X β. By hyothesis, there exists a fuzzy oen q-nbd µ of X such that f 1 (y ) µ Cl(µ) 1 X β. Since f is fuzzy weakly reclosed by Theorem 3.6, there exists a fuzzy reoen q-nbd v in Y with y v and f 1 (v) Cl(µ). Therefore, we obtain f 1 (v) q β and hence v q f(β), this shows that y Cl(f(β)). Therefore, f(β) is fuzzy reclosed in Y and f is fuzzy reclosed. Theorem 3.10: (i) If f : (X, τ 1 ) is fuzzy reclosed and fuzzy contra-closed, then f is fuzzy weakly reclosed. (ii) If f : (X, τ 1 ) is fuzzy contra-oen, then f is fuzzy weakly reclosed. Proof: (i) Let β be a fuzzy closed subset of X. Since f is fuzzy reclosed Cl(Int(f(β))) f(β) and since f is fuzzy contra-closed f(β) is fuzzy oen. Therefore Cl(f(Int(β))) Cl(f(β)) Cl(Int(f(β))) f(β). (ii) Let β be a fuzzy closed subset of X. Then, Cl(f(Int(β))) f(int(β)) f(β). Theorem 3.11: If f : (X, τ 1 ) is fuzzy weakly reclosed, then for every fuzzy subset β in Y and every fuzzy oen set λ in X with f 1 (β) λ, there exists a fuzzy reclosed set δ in Y such that β δ and f 1 (δ) Cl(λ).

10 46 International Journal of General Toology Proof: Let β be a fuzzy subset of Y and let λ be a fuzzy oen subset of X with f 1 (β) λ. Put δ = Cl(f(Int(Cl(λ))), then δ is a fuzzy reclosed set of Y such that β δ since β f(λ) f(int(cl(λ)) Cl(f(Int(Cl(λ))) = δ. And since f is fuzzy weakly reclosed, f 1 (δ) τ Cl(λ) (Theorem 3.5). Corollary 3.12: If f : (X, τ 1 ) is fuzzy weakly reclosed, then for every fuzzy oint y in Y and every fuzzy oen set λ in X with f 1 (y ) λ, there exists a fuzzy reclosed set δ in Y containing y such that f 1 (δ) Cl(λ). A fuzzy set β in a fts X is fuzzy θ-comact if for each cover Ω of β by fuzzy oen q-nbd µ in X, there is a finite family µ 1,..., µ n in Ω such that β Int( {Cl(µ i ) : i = 1, 2,..., n}). Theorem 3.13: If f : (X, τ 1 ) is fuzzy weakly reclosed with all fibers fuzzy θ-closed, then f(β) is fuzzy reclosed for each fuzzy θ-comact β in X. Proof: Let β be fuzzy θ-comact and let y 1 Y f(β). Then f 1 (y ) qβ and for each x 1 β there is a fuzzy oen q-nbd µ x containing x in X and Cl()() µ x ṗ q f y Clearly Ω = { µ x : x β} is a fuzzy oen q-nbd cover of β and since β is fuzzy θ-comact, there is a finite family { µ x,..., µ } 1 x n in Ω such that β Int(λ), where λ = {Cl( µ i x i ) : i = 1,..., n}. Since f is fuzzy weakly reclosed by Theorem 3.6 there exists a fuzzy reoen q-nbd δ in Y with f 1 (y ) f 1 (δ) Cl(1 X λ) = 1 X Int(λ) 1 X β. Therefore y δ and δqf (). β Thus y 1 Y Cl(f(β)). This shows that f(β) is fuzzy reclosed. Two non emty fuzzy subsets λ and β in X are strongly fuzzy searated if there exist fuzzy oen sets µ and v in X with λ µ and β v and Cl(µ) qcl(v). If λ and β are fuzzy singleton sets we may seak of fuzzy oints being strongly fuzzy searated. We will use the fact that in a fuzzy normal sace, disjoint fuzzy closed sets are strongly fuzzy searated. Recall that sace X is said to be fuzzy re-hausdorff or in short fuzzy re-t 2 [17] if for every air of fuzzy oints x and x q with different suorts, there exist two fuzzy reoen c sets λ and β such that x λ x and x β x and λqβ. c q q Theorem 3.14: If f : (X, τ 1 ) is a fuzzy weakly reclosed surjection and all airs of disjoint fibers are strongly fuzzy searated, then Y is fuzzy re-t 2. Proof: Let y and y q be two fuzzy oints in Y. Let µ and v be fuzzy oen sets in X such that f 1 (y ) µ and f 1 (y q ) µ resectively with Cl(µ) q Cl(v). By fuzzy weak reclosedness (Theorem 2.5 (iv)) there are fuzzy reoen sets λ and β in Y such that y λ and y q β, f 1 (λ) Cl(µ) and f 1 (β) Cl(v). Therefore λ q β, because Cl(µ) q Cl(v) and f surjective. Then Y is fuzzy re-t 2. Corollary 3.15: If f : (X, τ 1 ) is a weakly fuzzy reclosed surjection with all fuzzy fibers closed and X is fuzzy normal, then Y is a fuzzy re T 2 sace. Definition 3.16: A family {λ α : α Ω} of fuzzy oen subsets (res. fuzzy reclosed subsets) of a fuzzy toological sace (X, τ) is a fuzzy oen cover (res. fuzzy reclosed cover) if {λ α : α Ω} = X. An fts X is said to be fuzzy almost comact [7, 5] (res. fuzzy -comact) if every fuzzy oen cover (res. fuzzy reclosed cover) contains a finite µ x

11 Weakly Preoen and Weakly Preclosed Functions in Fuzzy Toology* 47 subfamily { λ α i : i = 1, 2,..., n} such that X = Cl (). λα i n i A fuzzy subset λ of a fts X is fuzzy almost comact relative to X (res. fuzzy -comact relative to X) if every cover of λ by fuzzy oen (res. fuzzy reclosed) sets of X has a finite subfamily whose fuzzy closures cover X. Recall that a fts (X, τ) is said to be fuzzy extremally disconnected if the closure of every fuzzy oen set of X is fuzzy oen in X [6]. A fts (X, τ) satisfies the roerty (o) if the finite intersection of reoen sets is reoen. Lemma 3.17: A function f : (X, τ 1 ) is fuzzy oen if and only if for each fuzzy subset β of Y, f 1 (Cl(β)) Cl(f 1 (β)) [18]. Theorem 3.18: Let X be an fuzzy extremally disconnected sace that satisfies the roerty (o). Let f : (X, τ 1 ) be a fuzzy oen and fuzzy weakly reclosed function which is one-one and such that f 1 (y ) is fuzzy almost comact relative to X for each fuzzy oint y in Y: If λ is fuzzy -comact relative to Y. Then f 1 (λ) is fuzzy almost comact. Proof: Let { vβ : β I}, I being the index set be a fuzzy oen cover of f 1 (λ). Then for each y λ f(x), f 1 (y ) {Cl(v β ) : β I(y )} = δ y for some finite subfamily I(y ) of I. Since X is fuzzy extremally disconnected each Cl(v β ) is fuzzy oen, hence δ y is fuzzy oen in X. So by Corollary 3.12, there exists a fuzzy reclosed set µ y containing y such that f 1 ( µ y ) Cl( δ y ). Then, { µ :()} {1()} y y λ f x Y f X is a fuzzy reclosed cover of λ, λ {Cl( µ y ) : y K} {Cl(1 Y f(x))} for some finite fuzzy subset K of λ f(x). Hence and by Lemma 3.17, f 1 (λ) {f 1 (Cl( µ y ) : y K} {f 1 (Cl(1 Y f(x))} {Cl(f 1 ( µ y ) : y K} {Cl(f 1 (1 Y f(x)))} {Cl(f 1 ( µ y )) : y K}, so f 1 (λ) {Cl(v β ) : β I(y ), y K}. Therefore f 1 (λ) is fuzzy almost comact. Corollary 3.19: Let f : (X, τ 1 ) be as in Theorem If Y is fuzzy -comact, then X is fuzzy almost comact. REFERENCES [1] K.K. Azad, On Fuzzy Semicontinuity, Fuzzy Almost Continuity and Fuzzy Weakly Continuity, J. Math. Anal. Al., 82, (1981), [2] R.N. Bhoumik, and A. Mukherjee, Fuzzy Weakly Comletely Continuous Functions, Fuzzy Sets and Systems, 55, (1993), [3] A.S. Bin Shahna, On Fuzzy Strong Semicontinuity and Fuzzy Precontinuity, Fuzzy Sets and Systems, 44, (1991), [4] C.L. Chang, Fuzzy Toological Saces, J. Math. Anal. Al., 24, (1968), [5] A. Di Concilio, and G. Gerla, Almost Comact in Fuzzy Toological Saces, Fuzzy Sets and Systems, 13, (1984), [6] B. Ghosh, Fuzzy Extremally Disconnected Saces, Fuzzy Sets and Systems, 46, (1992), [7] A. Haydar Es, Almost Comactness and Near Comactness in Fuzzy Toological Saces, Fuzzy Sets and Systems, 22, (1987),

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