() CUBO A Mathematical Journal Vol.13, No¯ 03, (49–56). October 2011 On Strongly Fβp-irresolute Mappings Ratnesh Kumar Saraf Department of Mathematics, Government Kamla Nehru, Girls College DAMOH (M.P.)-470661, India. and Miguel Caldas Departamento de Matemática Aplicada, Universidade Federal Fluminense Rua Mário Santos Braga s/n0,CEP: 24020-140, Niteroi-RJ,Brasil. email: gmamccs@vm.uff.br ABSTRACT In this paper, we introduce a new class of mappings called strongly Fβp-irresolute mappings between fuzzy topological spaces. We obtain several characterizations of this class and study its properties and investigate the relationship with the known mappings. RESUMEN En este trabajo presentamos una nueva clase de funciones llamadas funciones fuerte- mente Fβp-irresolute entre espacios topológicos difusos. Obtenemos varias caracteriza- ciones de esta clase, estudiamos sus propiedades e investigamos la relación con funciones conocidas. Keywords: Fuzzy topological spaces, fuzzy β-open sets, fuzzy β-preirresolute maps, strongly fuzzy β-preirresolute maps. 50 Ratnesh Kumar Saraf and Miguel Caldas CUBO 13, 3 (2011) Mathematics Subject Classification: 54C10, 54D10. 1 Introduction and preliminaries. The concept fuzzy has invaded almost all branches of mathematics with the introduction of fuzzy sets by Zadeh [23] of 1965. The theory of fuzzy topological spaces was introduced and developed by Chang [6] and since then various notions in classical topology have been extended to fuzzy topological spaces. Recently Professor El-Naschie has been shown in [7] and [8] that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and ε∞ theory. Thus our motivation in this paper is to define strongly fuzzy β-preirresolute (in short St-Fβp-irresolute) mappings and investigate its properties. The new defined class of map- ping is stronger that M-fuzzy β-continuous mappings and is a generalization of St-Fαp-irresolute mappings. Throughout this paper (X, τ), (Y, σ) and (Z, γ) (or simply X, Y and Z) represent non-empty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned. The fuzzy set A of X is called fuzzy α-open (Fα-open) [5] (resp. fuzzy preopen (Fp-open) [5], fuzzy β-open (Fβ-open) [2]) if A ≤ Int(Cl(Int(A)) (resp. A ≤ Int(Cl(A)), A ≤ Cl(Int(Cl(A))), where Cl(A) and Int(A) denote the closure of A and the interior of A respectively. The fuzzy subset B of X is said to be fuzzy α-closed (Fα-closed) (resp. fuzzy preclosed (Fp-closed), fuzzy β-closed (Fβ-closed)) if, its complement Bc is fuzzy Fα-open (resp. Fp-open, Fβ-open) in X. By FαO(X), FPO(X) and FβO(X) (resp. FαC(X), FPC(X), and FβC(X)) we denote the family of all Fα-open, Fp-open and Fβ-open (resp. Fα-closed, Fp-closed and Fβ-closed) sets of X. The intersection of all fuzzy β-closed sets containing A is called the β-closure of A and is denoted by βCl(A). The fuzzy β-interior [2] of A denoted by β-Int(A), is defined by the union of all fuzzy β-open sets of X contained in A. A mapping f : X → Y is said to be: (i) fuzzy completely weakly preirresolute [11] (resp. Fαp-irresolute [5], M-fuzzy precontinuous [3], Fβp-irresolute [17]) if, f−1(V) is fuzzy open (resp. Fα-open), Fp-open, Fβ-open) in X for every Fp-open set V of Y. (ii) strongly M-fuzzy β-continuous [16] (resp. M-fuzzy β-continuous [15], St-Fαp-irresolute [16]) if, f−1(V) is fuzzy open (resp. Fβ-open, Fα-open) in X for every Fβ-open set V of Y. (iii) fuzzy strongly continuous [12] if, f−1(V) is fuzzy clopen in X for every fuzzy subset V of Y. A fuzzy point in X with support x ∈ X and value p (0 < p ≤ 1) is denoted by xp. The fuzzy point xp is said to be quasi-coincident (shorty: q-coincident) with a fuzzy set A of X denoted by xpqA if p + A(x) > 1. Two fuzzy sets A and B are said to be quasi-coincident denoted by AqB, if there exists x ∈ X such that A(x) + B(x) > 1 [14] and by − q we denote ”is not q-coincident”. It is known [14] that A ≤ B if and only if Aq(1 − B). CUBO 13, 3 (2011) On Strongly Fβp-irresolute Mappings 51 Two non empty fuzzy subsets A and E are said to be fuzzy β-separated if there exist two fuzzy β-open subsets G and H such that A ≤ G, E ≤ H, A − qH and E − qG. A fuzzy subset which cannot be expressed as the union of two fuzzy β-separated subsets is said to be fuzzy β-connected sets. Lemma 1.1. [22] Let f : X → Y be a mapping and xp be a fuzzy point of X. Then: (1) f(xp)qB ⇒ xpqf−1(B), for every fuzzy set B of Y. (2) xpqA ⇒ f(xp)qf(A), for every fuzzy set A of X. 2 St-Fβp-irresolute mappings. Definition 2.1. A mapping f : X → Y is said to be strongly fuzzy β- preirresolute (briefly St-Fβp- irresolute) if, f−1(V) is fuzzy preopen in X for every Fβ-open set V of Y. From the definitions stated, we have the following diagram: A → B → C → D ↓ ↓ ↓ ↓ E → F → G → H Where: A = St-MFβ-continuous; B = St-Fαp-irresolute; C = St- Fβp-irresolute; D = MFβ- continuous; E = Fuzzy completely weakly preirresolute; F= Fαp-irrsesolute; G = MFp-continuous; H = Fβp-irresolute. Remark 2.1. However, converses of the above implications are not true in general, by [12, 16, 17] and the followings examples: (i) Fαp-irrsesolute mapping does not imply fuzzy completely weakly preirresolute: Let X = {a, b} and Y = {x, y}. Define fuzzy sets A(a) = 0.6, A(b) = 0.5; B(a) = 0, B(b) = 0.8; H(x) = 0.5, H(y) = 0.5; E(x) = 0.7, E(y) = 0.8. Let τ = {0, A, 1}, Γ = {0, B, 1}; σ = {0, H, 1} and υ = {0, E, 1}. The mapping f : (X, τ) → (Y, σ) defined by f(a) = x , f(b) = y is fuzzy α-preirresolute but not fuzzy completely weakly preirresolute, because Z(x) = 0.7, Z(y) = 0.7 are fuzzy preopen in (Y, σ) but f−1(Z) is not fuzzy open in X. (ii) Fuzzy completely weakly preirresolute mapping does not imply MFβ-continuous, see [[18], Ex- ample 3.2]. (iii) MFβ-continuous mapping does not imply MFp-continuous, see [[19], Example 3.1]. (iv) St- Fβp-irresolute mapping does not imply Fαp-irrsesolute, see [[19], Example 3.2]. 52 Ratnesh Kumar Saraf and Miguel Caldas CUBO 13, 3 (2011) (v) St- Fαp-irresolute mapping does not imply fuzzy completely weakly preirresolute, see [[16], Example 3.1]. Theorem 2.1. For a mapping f : X → Y, the following are equivalent: (1) f is St-Fβp-irresolute; (2) For every fuzzy point xt in X and every Fβ-open set V of Y containing f(xt), there exist a Fp-open set U of X containing xt such that f(U) ≤ V; (3) For every fuzzy point xt in X and every Fβ-open set V of Y containing f(xt), there exist a Fp-open set U of X containing xt such that xt ∈ U ≤ f −1(V); (4) For every fuzzy point xt in X, the inverse image of each β-neighbourhood of f(xt) is a preneigh- bourhood of xt; (5) For every fuzzy point xt in X and each β-neighbourhood E of f(xt), there exists an preneigh- bourhood A of xt such that f(A) ≤ E; (6) f−1(V) ≤ Int(Cl(f−1(V))) for every V ∈ FβO(Y); (7) f−1(H) ∈ FPC(X) for every H ∈ FβC(Y) ; (8) Cl(Int(f−1(E))) ≤ f−1(βCl(E)) for every fuzzy subset E of Y; (9) f(Cl(Int(A))) ≤ βCl(f(A))) for every fuzzy subset A of X. Proof. (1) ⇔ (2) ⇔ (3); (4) ⇒ (5): Obvious (2) ⇒ (6): Let V ∈ FβO(Y) and xt ∈ f−1(V). By (2), there exists U ∈ FPO(X) containing xt such that f(U) ≤ V. Thus we have xt ∈ U ≤ Int(Cl(U)) ≤ Int(Cl(f −1(V))) and hence f−1(V) ≤ Int(Cl(f−1(V))). (6) ⇒ (7): Let H ∈ FβC(Y). Set V = Y − H, then V ∈ FβO(Y). By (6) we obtain f−1(V) ≤ Int(Cl(f−1(V))) and hence f−1(H) = X − f−1(Y − H) = X − f−1(V) ∈ FPC(X). (7) ⇒ (8): Let E be any fuzzy set of Y. Since βCl(E) ∈ FβC(Y), then f−1(βCl(E)) ∈ FPC(X) and hence Cl(Int(f−1(βCl(E)))) ≤ f−1(βCl(E)). Therefore we obtain Cl(Int(f−1(E))) ≤ f−1(βCl(E)). (8) ⇒ (9): Let A be any fuzzy set of X. by (8), we have Cl(Int(A)) ≤ Cl(Int(f−1(f(A)))) ≤ f−1(βCl(f(A))) and hence f(Cl(Int(A))) ≤ βCl(f(A)). (9) ⇒ (1): Let V ∈ FβO(Y). Since f−1(Y − V) = X − f−1(V) is a fuzzy set of X and by (9), we obtain f(Cl(Int(f−1(Y − V)))) ≤ βCl(f(f−1(Y − V))) ≤ βCl(Y − V) = Y − βInt(V) = Y − V and hence X − Int(Cl(f−1(V))) = Cl(Int(X − f−1(V)))) = Cl(Int(f−1(Y − V))) ≤ f−1(f(Cl(Int(f−1(Y − V))))) ≤ f−1(Y − V) = X − f−1(V). Therefore, we have f−1(V) ≤ Int(Cl(f−1(V))) and hence f−1(V) ∈ FPO(X). Thus, f is St-Fβp-irresolute. (1) ⇒ (4): Let xt be a fuzzy point in X and V be any β-neighbourhood of f(xt), then there exists G ∈ FβO(Y) such that, f(xt) ∈ G ≤ V. Now f −1(G) ∈ FPO(X) and xt ∈ f −1(G) ≤ f−1(V). Thus f−1(V) is an preneighbourhood of xt in X. (5) ⇒ (2): Let xt be a fuzzy point in X and V ∈ FβO(Y) such that f(xt) ∈ V. Then V is β- neighbourhood of f(xt), so there is a preneighbourhood A of xt such that xt ∈ A, and f(A) ≤ V. Hence there exists U ∈ FPO(X) such that xt ∈ U ≤ A, and so f(U) ≤ f(A) ≤ V. CUBO 13, 3 (2011) On Strongly Fβp-irresolute Mappings 53 Theorem 2.2. For a function f : X → Y, the following are equivalent: (1) f is St-Fβp-irresolute; (2) For each fuzzy point xt of X and every E ∈ FβO(Y) such that f(xt)qE, there exists A ∈ FPO(X) such that xtqA and f(A) ≤ E; (3) For every fuzzy point xt of X and every E ∈ FβO(Y) such that f(xt)qE, there exists A ∈ FPO(X) such that xtqA and A ≤ f −1(E). Proof. (1) ⇒ (2) Let xt be a fuzzy point in X and E ∈ FβO(Y) such that f(xt)qE. Then f−1(E) ∈ FPO(X), and xtqf −1(E) by Lemma 1.1. If we take A = f−1(E) then xtqA and f(A) = f(f−1(E)) ≤ E. (2) ⇒ (3) Let xt be a fuzzy point in X and E ∈ FβO(Y) such that f(xt)qE. Then by (2), there exists A ∈ FPO(X) such that xtqA and f(A) ≤ E. Hence we have xtqA and A ≤ f −1(f(A)) ≤ f−1(E). (3) ⇒ (1) Let E ∈ FβO(Y) and xt be a fuzzy point of X such that xt ∈ f−1(E). Then f(xt) ∈ E. Choose the fuzzy point xct (x) = 1 − xt(x). Then f(x c t )qE. And so by (3), there exists A ∈ FPO(X) such that xct qA and f(A) ≤ E. Now x c t qA implies x c t (x) + A(X) = 1 − xt(x) + A(x) > 1. It follows that xt ∈ A. Thus xt ∈ A ≤ f −1(E). Hence f−1(E) ∈ FPO(X). Lemma 2.1. [1] Let g : X → X × Y be the graph of a mapping f : X → Y. If A is a fuzzy set of X and B is a fuzzy of Y, then g−1(A × B) = A ∩ f−1(B) Theorem 2.3. A mapping f : X → Y is St-Fβp-irresolute if the graph mapping g : X → X × Y, is St-Fβp-irresolute. Proof. Let V be any Fβ-open set of Y, then by Lemma 2.1, f−1(V) = 1X ∩ f −1(V) = g−1(1x × V). Since V is Fβ-open in Y, 1X ×V is Fβ-open in X×Y. Since g is St-Fβp-irresolute g −1(1x ×V) ∈ FpO(X) and hence f−1(V) is Fp-open in X and consequently f is St-Fβp-irresolute. Theorem 2.4. If f : X → Y is St-Fβp-irresolute and g : Y → Z is M-fuzzy β-continuous, then g ◦ f : X → Z is St-Fβp-irresolute. Proof. Straightforward. Corollary 2.1. The composition of two St-Fβp-irresolute mapping is St-Fβp-irresolute. Corollary 2.2. If f : X → Y is fuzzy strongly continuous and g : Y → Z is St-Fβp-irresolute, then g ◦ f : X → Z is St-Fβp-irresolute. Proof. Obvious. Theorem 2.5. If f : X → Y is M-fuzzy β-continuous and g : Y → Z is St-Fβp-irresolute, then g ◦ f : X → Z is St-Fβp-irresolute. Theorem 2.6. Let {Xi : i ∈ Ω} be any family of fuzzy topological spaces. If f : X → ∏ Xi is St-Fβp-irresolute, then for each i ∈ Ω, fi : X → Xi is St-Fβp-irresolute. 54 Ratnesh Kumar Saraf and Miguel Caldas CUBO 13, 3 (2011) Proof. Let Pri be the projection of ∏ Xi onto Xi, we know that if a mapping is fuzzy continuous and fuzzy open, then it is M-fuzzy β-continuous [21]. So the mapping Pri is M-fuzzy β-continuous. Now for each i ∈ Ω, fi = Pri ◦ f : X → Xi. It follows from Theorem 2.1 that fi is St-Fβp-irresolute since f is St-Fβp-irresolute. 3 Preservation of some fuzzy topological structure. In this section preservation of some fuzzy topological structure under the St-Fβp-irresolute map- ping are studied. Let us recall the definition: A space X is said to be fuzzy β-compact [4] if for every Fβ-open cover of X has a finite subcover, and X is fuzzy strongly compact [13] if for every Fp-open cover of X has a finite subcover. Theorem 3.1. Every surjective St-Fβp-irresolute image of a fuzzy strongly compact space is fuzzy β-compact. Proof. Let f : X → Y be St-Fβp-irresolute mapping of a fuzzy strongly compact space X onto a space Y. Let {Gi : i ∈ Ω} be any Fβ-open cover of Y. Then {f −1(Gi) : i ∈ Ω} is a Fp-open cover of X. Since X is fuzzy strongly compact, there exist a finite subfamily {f−1(Gij ) : j = 1, 2, ..., n} of {f−1(Gi) : i ∈ Ω} which covers X. It follows that {Gij : j = 1, 2, ..., n} is a finite subfamily of {Gi : i ∈ Ω} which covers Y. Hence Y is Fβ-compact. Theorem 3.2. Let f : X → Y be a St-Fβp-irresolute mapping. If A is a Fβ-connected subset of X, then f(A) is also Fβ-connected in Y. Proof. Suppose f(A) is not Fβ-connected in Y. Then there exist Fβ-separated subset G and H in Y, such that f(A) = G ∪ H. Since G and H are Fβ-separated, there exist two Fβ-open, subset U and V such that G ≤ U, H ≤ V, G − qV and H − qU. Now f being St-Fβp-irresolute so f−1(G) and f−1(H) are Fp-open in X. Thus Fβ-open in X and A = f−1(f(A)) = f−1(G ∪ H) = f−1(G) ∪ f−1(H). 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