Fuzzy Soft Topology. G. Kalpana and C. Kalaivani Department of Mathematics, SSN College of Engineering, Kalavakkam , Chennai, India.

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1 International Journal of Engineering Studies. ISSN Volume 9, Number 1 (2017), pp Research India Publications Fuzzy Soft Topology G. Kalpana and C. Kalaivani Department of Mathematics, SSN College of Engineering, Kalavakkam , Chennai, India. Abstract In the present paper we introduce the concept of fuzzy soft topological space in Šostak sense. Then the notions of fuzzy soft r-closure and r-interior are introduced and their basic properties are investigated. Furthermore, fuzzy soft r -continuous mapping, fuzzy soft r -open, fuzzy soft r -closed mappings and fuzzy soft r -homeomorphism for fuzzy soft topological spaces are given and structural characteristics are discussed. 1. INTRODUCTION The notion of a fuzzy set was introduced by Zadeh [13] in his classical paper of In 1968, Chang [2] gave the definition of fuzzy topology, which is family of fuzzy sets satisfying the three classical axioms. Later, fuzzy topological space is generalized in different ways, one of which is developed by Šostak [10]. Then plenty of works on Šostak fuzzy topological spaces have been done in order to extend various concepts in classical topology. In 1999, Molodtsov [7] introduced the concept of a soft set and started to develop the basics of the corresponding theory as a new approach for modeling uncertainties. Maji et al. [5,6] presented some new definitions on soft sets and discussed in detail the application of soft set theory in decision making problems. Shabir and Naz [9] introduced the concept of soft topological space and studied neighborhoods and separation axioms. Tanay and Kandemir [11] introduced the definition of fuzzy soft topology over a subset of the initial universe set. Later, Roy

2 46 G. Kalpana and C. Kalaivani and Samanta [8] gave the definition of fuzzy soft topology over the initial universe set. It was further studied by Varol and Aygu n [12] and Cetkin and Aygun [3] etc. The aim of the paper is to introduced fuzzy soft topology in Šostak sense. Also, we introduce the notion of fuzzy soft r -closure, fuzzy soft r -interior and discussed some of its basic properties. Furthermore, we also define fuzzy soft r -continuity, fuzzy soft r -open mapping, fuzzy soft r -closed mapping and fuzzy soft r - homeomorphism and investigate some characterization. 2. PRELIMINARIES Throughout this paper, X refers to an initial universe, E is the set of all parameters for X, I = [0,1], I 0 = (0,1] and I X is the set of all fuzzy sets on X (X, E) denotes the collection of all fuzzy soft sets on X and is called a fuzzy soft universe. Definition 2.1. [4] f A is called a fuzzy soft set on X, where f is a mapping from the parameter set, E into I X, i.e., f A (e) is a fuzzy set on X, for each e A E and f A (e) = 0 X if e A, where 0 X denotes empty fuzzy set on X. Definition 2.2. [4] Two fuzzy soft sets f A and g B on X we say that f A is called fuzzy soft subset of g B and write f A g B if f A (e) g B (e), for every e E. Definition 2.3. [4] Two fuzzy soft sets f A and g B on X are called equal if f A g B and g B f A. Definition 2.4. [4] Let f A, g B (X, E). Then the union of f A and g B is also a fuzzy soft set h C, defined by h C (e) = f A (e) g B (e) for all e E, where C = A B. Here we write h C = f A g B. Definition 2.5. [4] Let f A, g B (X, E). Then the interasection of f A and g B is also a fuzzy soft set h C, defined by h C (e) = f A (e) g B (e) for all e E, where C = A B. Here we write h C = f A g B. Definition 2.6. [4] A fuzzy soft set f E on X is called a null fuzzy soft set denoted by 0 E if f E (e) = 0 X for each e E. Definition 2.7. [4] A fuzzy soft set f E on X is called a absolute fuzzy soft set denoted by 1 E if f E (e) = 1 X for each e E. Definition 2.8. [4] Let f A (X, E). Then the complement of f A is denoted by f A c and is defined by f A c (e) = 1 f A (e) for each e E.

3 Fuzzy Soft Topology 47 Lemma 2.9. [1] Let be an index set and f A, g B, h C, (f A ) i, (g B ) i (X, E), i, then we have the following properties: (1) f A f A = f A, f A f A = f A. (2) f A g B = g B f A, f A g B = g B f A. (3) f A (g B h C ) = (f A g B ) h C, f A (g B h C ) = (f A g B ) h C. (4) f A = f A (g B h C ), f A = f A (g B h C ). (5) f A ( i (g B ) i ) = i (f A (g B ) i ). (6) f A ( i (g B ) i ) = i (f A (g B ) i ). (7) 0 E f A 1 E. (8) (f A c ) c = f A. (9) ( i (f A ) i ) c = i (f A ) i c. (10) ( i (f A ) i ) c = i (f A ) i c. (11) If f A g B, then g A c f A c. Definition [1] Let φ: X Y and ψ: E F be two mappings, where E and F are parameter sets for the crisp sets X and Y, respectively. Then φ ψ is called a fuzzy soft mapping from (X, E) into (Y, F) and denoted by φ ψ : (X, E) (Y, F). Definition [1] Let f A and g B be two fuzzy soft sets over X and Y, respectively and let φ ψ be a fuzzy soft mapping from (X, E) into (Y, F). (1) The image of f A under the fuzzy soft mapping φ ψ, denoted by φ ψ (f A ) and is defined as, φ ψ (f A ) k (y) = { φ(x)=y ψ(e)=k f A (e)(x), if φ 1 (y), ψ 1 (k) ; 0, otherwise, for all k F, for all y Y. (2) The preimage of g B under the fuzzy soft mapping φ ψ, denoted by φ ψ 1 (g B ) and is defined as, φ ψ 1 (g B )(e)(x) = g B (ψ(e))(φ(x)), for all e E, for all x.

4 48 G. Kalpana and C. Kalaivani Definition [1] If φ and ψ are injective (surjective) then the fuzzy soft mapping φ ψ is injective (surjective). If φ ψ is both injective and surjective, then it is called bijective. Lemma [1] Let X and Y be two Universes, f A, (f A ) i (X, E), g B, (g B ) i (Y, F), for all i where is an index set, and φ ψ is a soft mapping from (X, E) into (Y, F). Then, (1) If (f A ) 1 (f A ) 2, then φ ψ ((f A ) 1 ) φ ψ ((f A ) 2 )). (2) If (g B ) 1 (g B ) 2, then φ ψ 1 ((g B ) 1) φ ψ 1 (g B ) 2 ). (3) f A φ ψ 1 (φ ψ (f A )), the equality holds if φ ψ is injective. (4) φ ψ (φ ψ 1 (g B )) g B, the equality holds if φ ψ is surjective. (5) φ ψ ( i (f A ) i ) = i φ ψ ((f A ) i ). (6) φ ψ ( i (f A ) i ) i φ ψ ((f A ) i ), the equality holds if φ ψ is injective. (7) φ ψ 1 ( i (g B ) i ) = i φ ψ 1 ((g B ) i ). (8) φ ψ 1 ( i (g B ) i )= i φ ψ 1 ((g B ) i ). (9) φ ψ 1 (g B c ) = (φ ψ 1 (g B )) c. (10) φ ψ 1 (0 F ) = 0 E, φ ψ 1 (1 F ) = 1 E. (11) φ ψ (1 E ) = 1 F if φ ψ is surjective and φ ψ (0 E ) = 0 F. 3. FUZZY SOFT r-closure AND r-interior In this section, we generalized the concept of fuzzy topological space (X, τ, E) in the sense of Šostak. Here, τ: (X, E) I is a mapping which assigns to every fuzzy soft set of X the real number. The value of τ(f A ) is interpreted as the degree of openness of a fuzzy soft set f A. Definition 3.1. A fuzzy soft topology is a map τ: (X, E) I, satisfying the following three axioms: (1) τ(0 E ) = τ(1 E ) = 1, (2) τ(f A g B ) τ(f A ) τ(g B ), for all f A, g B (X, E), (3) τ( i (f A ) i ) i J τ((f A ) i ), for all ((f A ) i ) i J (X, E).

5 Fuzzy Soft Topology 49 Then (X, τ, E) is called a fuzzy soft topological space. Definition 3.2. Let (X, τ, E) be a fuzzy soft topological space and η: (X, E) I defined by η(f A ) = τ(f c A ), satisfying the following three axioms: (1) η(0 E ) = η(1 E ) = 1, (2) η(f A g B ) η(f A ) τ(g B ), for all f A, g B (X, E), (3) η( i (f A ) i ) i J η((f A ) i ), for all ((f A ) i ) i J (X, E). Then (X, τ, E) is called a fuzzy soft cotopological space. Counterexample 3.3. Let X = {x, y}, E = {e 1, e 2, e 3 }, A = {e 1, e 2, }, B = {e 2, e 3 }, C = {e 2 }. We define f A, g B, h C, i E (X, E) by f A = {f A (e 1 ) [0.6,0.5] [x,y], f A (e 2 ) [0.5,0.4] [x,y] }, g B = {g B (e 2 ) [0.5,0.4] [x,y], g B (e 3 ) [0.7,0.2] [x,y] }, h C = { h C (e 2 ) [0.5,0.4] [x,y] } and i E = {i E (e 1 ) [0.5,0.4] [x,y], i E (e 2 ) [0.5,0.4] [x,y], i E (e 3 ) [0.7,0.2] [x,y] }. We define τ: (X, E) I by (1) Clearly, τ(0 E ) = τ(1 E ) = 1, 1, U = 0 E or 1 E, 0.6, U = f A, 0.5, U = g B, τ(u) = 0.7, U = h C, 0.5, U = i E, 0, { otherwise. (2) Clearly, we have f A g B = h C, f A h C = h C, g B h C = h C, f A i E = f A, g B i E = g B, f A, g B (X, E), h C i E = h C. Hence, τ ( f A g B ) τ(f A ) τ(g B ), for all (3) Clearly we verified that τ( i (f A ) i ) i J τ((f A ) i ), for all ((f A ) i ) i J (X, E). Hence, (X, τ, E) is called a fuzzy soft topological space. Definition 3.4. Let (X, τ, E) be a fuzzy soft topological space and f A (X, E). Then f A is called fuzzy soft r-open if τ(f A ) r, where r I 0. Definition 3.5. Let (X, τ, E) be a fuzzy soft topological space and f A (X, E). Then f A is called fuzzy soft r-closed if τ(f A c ) r, where r I 0.

6 50 G. Kalpana and C. Kalaivani Definition 3.6. Let (X, τ, E) be a fuzzy soft topological space and f A (X, E), r I 0. Then, fuzzy soft r -closure of f A is denoted by Cl(f A, r) and is defined by Cl(f A, r) = { f C : τ(f c C ) r, f A f C }. Clearly, Cl(f A, r) is the smallest fuzzy soft r-closed set which contains f A. Definition 3.7. Let (X, τ, E) be a fuzzy soft topological space and f A (X, E), r I 0. Then, fuzzy soft r -interior of f A is denoted by Int(f A, r) and is defined by Int(f A, r) = { f B : τ(f B ) r, f B f A }. Clearly, Int(f A, r) is the largest fuzzy soft r-open set which contained in f A. Theorem 3.8. Let (X, τ, E) be a fuzzy soft topological space. For each f A, g B (X, E) and r I 0. Then, (1) Cl(0 E, r) = 0 E. (2) f A Cl(f A, r). (3) Cl(Cl(f A,, r), r) = Cl(f A, r). (4) If f A g B, then Cl(f A, r) Cl(g B, r). (5) f A is fuzzy soft r -closed if and only if f A = Cl(f A, r). (6) Cl(f A g B, r) = Cl(f A, r) Cl(g B, r). (7) Cl(f A g B, r) Cl(f A, r) Cl(g B, r). Proof. (1) Clearly, 0 E Cl(0 E, r). Since, 0 E is fuzzy soft r-closed set and Cl(0 E, r) is the smallest r-closed set containing 0 E, Cl(0 E, r) 0 E. Therefore, Cl(0 E, r) = 0 E. (2) By the definition of fuzzy soft r -closure, f A Cl(f A, r). (3) By (2), Cl(f A, r) Cl(Cl(f A, r), r). Here Cl(f A, r) is fuzzy soft r-closed set. Since, Cl(Cl(f A, r), r) is the smallest fuzzy soft r -closed set containing Cl(f A, r), Cl(Cl(f A, r), r) Cl(f A, r).therefore, Cl(Cl(f A,, r), r) = Cl(f A, r). (4) Since f A g B, we have { f C : τ(f C c ) r, f A f C } { f C : τ(f C c ) r, g B f C }. Therefore, Cl(f A, r) = { f C : τ(f C c ) r, f A f C } { f C : τ(f C c ) r, g B f C } = Cl(g B, r). (5) Assume that f A is fuzzy soft r-closed set. (2), we have f A Cl(f A, r). Since f A is fuzzy soft r-closed set and Cl(f A, r) is the smallest fuzzy soft r-closed set containing

7 Fuzzy Soft Topology 51 f A, Cl(f A, r) f A. Hence, Cl(f A, r)= f A. Conversely, assume that Cl(f A, r)= f A. Since, Cl(f A, r) is the smallest fuzzy soft r - closed set. Hence, f A is fuzzy soft r -closed set. (6) Clearly, f A g B Cl( f A, r) Cl(g B, r). Since, Cl(f A g B, r)is the smallest fuzzy soft r -closed containing f A g B, Cl(f A g B, r) Cl(f A, r) Cl(g B, r). Clearly, Cl(f A, r) Cl(f A g B, r) and Cl(g B, r) Cl(f A g B, r). Therefore, Cl(f A, r) Cl(g B, r) Cl(f A g B, r). Hence, Cl(f A g B, r) = Cl(f A, r) Cl(g B, r). (7) Clearly, f A g B f A and f A g B g B. By (4), Cl(f A g B, r) Cl(f A, r) and Cl(f A g B, r) Cl(g B, r). Therefore, Cl(f A g B, r) Cl(f A, r) Cl(g B, r). Theorem 3.9. Let (X, τ, E) be a fuzzy soft topological space. For each f A, g B (X, E) and r I 0. Then, (1) Int(1 E, r) = 1 E. (2) Int(f A, r) f A. (3) Int(Int(f A,, r), r) = Int(f A, r). (4) If f A g B, then Int(f A, r) Int(g B, r). (5) f A is fuzzy soft r -open if and only if f A = Int(f A, r). (6) Int(f A g B, r) = Int(f A, r) Int(g B, r). (7) Int(f A, r) Int(g B, r) Int(f A g B, r). Proof. (1) Clearly, Int(1 E, r) 1 E. Since, 1 E is fuzzy soft r-open set and Int(1 E, r) is the largest r-open set contained in 1 E, 1 E Int(1 E, r). Therefore, Int(1 E, r) = 1 E. (2) By the definition of fuzzy soft r -interior, Int(f A, r) f A. (3) By (2), Int(Int(f A, r), r) Int(f A, r). Here Int(f A, r) is fuzzy soft r-open set. Since, Int(Int(f A, r), r) is the largest fuzzy soft r-open set contained in Int(f A, r), Int(f A, r) Int(Int(f A, r), r). Therefore, Int(Int(f A,, r), r) = Int(f A, r). (4) Assume that f A g B. Therefore, Int(f A, r) = { f C : τ(f C ) r, f C f A } { f C : τ(f C ) r, f C g B } = Int(g B, r).

8 52 G. Kalpana and C. Kalaivani (5) Assume that f A is fuzzy soft r-open set. By (2), we have Int(f A, r) f A. Since f A is fuzzy soft r-open set and Int(f A, r) is the largest fuzzy soft r-open set contained in f A, f A Int(f A, r). Hence, f A = Int(f A, r). Conversely, assume that f A = Int(f A, r). Since, Int(f A, r) is the largest fuzzy soft r-open set. Hence, f A is fuzzy soft r open set. (6) Clearly, Int( f A, r) Int(g B, r) f A g B. Since, Int(f A g B, r)is the largest fuzzy soft r-open contained in f A g B, Int(f A, r) Int(g B, r) Int(f A g B, r). Clearly, Int(f A g B, r) Int(f A, r) and Int(f A g B, r) Int(g B, r). Therefore, Int(f A g B, r) Int(f A, r) Int(g B, r). Hence, Int(f A g B, r) = Int(f A, r) Int(g B, r). (7) Clearly, f A f A g B and g B f A g B. By (4), Int(f A, r) Int(f A g B, r) and Int(g B, r) Int(f A g B, r). Therefore, Int(f A, r) Int(g B, r) Int(f A g B, r). Theorem Let (X, τ, E) be a fuzzy soft topological space. For each f A (X, E). Then, (1) (Int(f A, r)) c = Cl(f A c, r). (2) (Cl(f A, r)) c = Int(f A c, r). Proof. (1) Clearly, Int(f A, r) f A By Theorem 2.13(11), f c A (Int(f A, r)) c. Since (Int(f A, r)) c is fuzzy soft r-closed and by Theorem 3.8 (2) and (5), Cl(f c A, r) Cl((Int(f A, r)) c ) = (Int(f A, r)) c. Now, f c A Cl(f c A, r) By Theorem 2.13(11), (Cl(f c A, r)) c (f c A ) c = f A. Clearly (Cl(f c A, r)) c fuzzy soft r-open. Since Int(f A, r) is the largest fuzzy soft r-open set contained in f A, (Cl(f c A, r)) c Int(f A, r). By Theorem 2.13(11), (Int(f A, r)) c (Cl(f c A, r)) c = Cl(f c A, r). (2) Similar to (1). 4. FUZZY SOFT r-continuity Definition 4.1. Let φ ψ : (X, τ 1, E) : (Y, τ 2, F) be a fuzzy soft mapping and r I 0. If φ ψ is called fuzzy soft r-continuous if τ 1 (φ 1 ψ (g B )) r whenever g B (Y, F) and τ 2 (g B ) r. Definition 4.2. Let φ ψ : (X, τ 1, E) : (Y, τ 2, F) be a fuzzy soft mapping and r I 0.

9 Fuzzy Soft Topology 53 If φ ψ is called fuzzy soft r-open if τ 2 (φ ψ ( f A )) r whenever f A (X, E) and τ 1 ( f A ) r. Definition 4.3. Let φ ψ : (X, τ 1, E) : (Y, τ 2, F) be a fuzzy soft mapping and r I 0. If φ ψ is called fuzzy soft r-closed if η 2 (φ ψ ( f A )) r whenever f A (X, E) and η 1 ( f A ) r. Theorem 4.4. Let φ ψ : (X, τ 1, E) : (Y, τ 2, F) be a fuzzy soft mapping and r I 0. Then the following are equivalent: (1) φ ψ is fuzzy soft r-continuous. (2) η 1 (φ 1 ψ (g B )) r whenever g B (Y, F) and η 2 (g B ) r. (3) φ ψ (Cl(f A, r)) Cl(φ ψ (f A ), r), for all f A (X, E). (4) Cl (φ 1 ψ (g B ), r) φ 1 ψ (Cl(g B, r)) for all g B (Y, F). (5) φ 1 ψ (Int(g B, r)) Int (φ 1 ψ (g B ), r), for all g B (Y, F). Proof. (1) (2). Assume that φ ψ is fuzzy soft r-continuous. Let g B (Y, F) and η 2 (g B ) r. Clearly η 2 (g B ) = τ 2 (g B c ) r. Since φ ψ is fuzzy soft r -continuous, τ 1 (φ ψ 1 (g B c )) r. By Lemma 2.13(9), φ ψ 1 (g B c ) = (φ ψ 1 ( g B )) c. Hence, τ 1 ((φ ψ 1 (g B )) c ) r. Therefore, η 1 (φ ψ 1 (g B )) r. (2) (3). Assume η 1 (φ 1 ψ (g B )) r whenever g B (Y, F) and η 2 (g B ) r. Let f A (X, E). By Lemma 2.13(3) and by Theorem 3.8 (2), f A φ ψ 1 (φ ψ (f A )) φ ψ 1 (Cl(φ ψ (f A ), r)). Since Cl(φ ψ (f A ), r) is fuzzy soft r -closed set, η 2 (Cl(φ ψ (f A ), r)) r. By our assumption η 1 (φ ψ 1 (Cl(φ ψ (f A ), r))) r. Hence, φ 1 ψ (Cl(φ ψ (f A ), r)) is a fuzzy soft r-closed set such that f A φ 1 ψ (Cl(φ ψ (f A ), r)). Since Cl(f A, r) is the smallest fuzzy soft r-closed set containing f A, Cl(f A, r) φ 1 ψ (Cl(φ ψ (f A ), r)). By Lemma 2.13(4), (Cl(f A, r)) φ ψ (φ 1 ψ (Cl(φ ψ (f A ), r))) Cl(φ ψ (f A ), r). (3) (4). Assume that φ ψ (Cl(f A, r)) Cl(φ ψ (f A ), r), for all f A (X, E). Let gb

10 54 G. Kalpana and C. Kalaivani (Y, F). Then, 1 φψ (g B ) (X, E). By our assumption and by Lemma 2.13 (4), φ ψ (Cl(φ ψ 1 (g B ), r)) Cl(φ ψ (φ ψ 1 (g B )), r) Cl(g B, r). By Lemma 2.13 (3), Cl(φ ψ 1 (g B ), r) φ ψ 1 (φ ψ (Cl(φ ψ 1 (g B ), r)) φ ψ 1 (Cl(g B, r)). (4) (5). Assume that Cl (φ 1 ψ (g B ), r) φ 1 ψ (Cl(g B, r)) for all g B (Y, F). Let g B (Y, F). Clearly, g c B (Y, F). By our assumption, Cl (φ ψ 1 (g B c ), r) φ ψ 1 (Cl(g B c, r)). By Theorem 3.10 (1) and by Lemma 2.13 (9), (Int((φ ψ 1 (g B ), r)) c = Cl ((φ ψ 1 (g B )) c, r) Therefore, φ ψ 1 (Int(g B, r)) Int (φ ψ 1 (g B ), r). (5) (1). = Cl (φ ψ 1 (g B c ), r) φ ψ 1 (Cl(g B c, r)) = φ ψ 1 (Int(g B, r) c ) = (φ ψ 1 (Int(g B, r))) c. Assume that φ 1 ψ (Int(g B, r)) Int (φ 1 ψ (g B ), r), for all g B (Y, F). Let g B (Y, F) and τ 2 (g B ) r. Therefore, g B = Int(g B, r). By our assumption and by Theorem 3.9 (2), φ ψ 1 (g B ) = φ ψ 1 (Int(g B, r)) Int (φ ψ 1 (g B ), r) φ ψ 1 (g B ). Hence, φ ψ 1 (g B ) = Int (φ ψ 1 (g B ), r). By Theorem 3.9(5), (φ ψ 1 (g B ) is fuzzy soft r -open. Therefore, τ 1 (φ ψ 1 (g B )) r. Thus, φ ψ is fuzzy soft r-continuous. Theorem 4.5. Let φ ψ : (X, τ 1, E) : (Y, τ 2, F) be a fuzzy soft mapping and r I 0. Then, (1) φ ψ is fuzzy soft r-open mapping iff for each fuzzy soft set f A over X, φ ψ (Int(f A, r)) Int(φ ψ ( f A ), r). (2) φ ψ is fuzzy soft r-closed mapping iff for each fuzzy soft set f A over X,

11 Fuzzy Soft Topology 55 Cl(φ ψ ( f A ), r) φ ψ (Cl(f A, r)). Proof. (1) Let φ ψ is fuzzy soft r-open mapping and f A be a fuzzy soft set over X. Clearly, Int(f A, r) is fuzzy soft r -open set. Since φ ψ is fuzzy soft r -open and Int(f A, r) f A, φ ψ (Int(f A, r)) is fuzzy soft r- open set in Y and φ ψ (Int(f A, r)) φ ψ ( f A ). Since Int( φ ψ ( f A ), r) is the largest r -open set contained in φ ψ ( f A ), φ ψ (Int(f A, r)) Int( φ ψ ( f A ), r). Conversely τ 1 ( f A ) r, where f A (X, E). Hence, Int(fA, r) = f A. By our assumption and by Theorem 3.9 (2), φ ψ ( f A ) = φ ψ (Int(f A, r) Int( φ ψ ( f A ), r) φ ψ ( f A ). Hence, φ ψ ( f A ) = Int( φ ψ ( f A ), r). Therefore, τ 2 ( φ ψ ( f A ), r) r. Thus φ ψ is a fuzzy soft r -open. (2) Let φ ψ is fuzzy soft r-closed mapping and f A be a fuzzy soft set over X. Clearly, Cl(f A, r) is fuzzy soft r -closed set. Since φ ψ is fuzzy soft r -closed and f A Cl(f A, r), φ ψ (Cl(f A, r)) is fuzzy soft r - closed set in Y and φ ψ ( f A ) φ ψ (Cl(f A, r)). Since Cl( φ ψ ( f A ), r) is the smallest r-closed set containing φ ψ ( f A ), Cl(φ ψ ( f A ), r) φ ψ (Cl(f A, r)). Conversely f A is fuzzy r-closed set over X. By our assumption and by Theorem 3.8 (2) and (5), Cl( φ ψ ( f A ), r) φ ψ ( f A ) = φ ψ (Int(f A, r) Int( φ ψ ( f A ), r) φ ψ ( f A ). Hence, φ ψ ( f A ) = Int( φ ψ ( f A ), r). Therefore, τ 2 ( φ ψ ( f A ), r) r. Thus φ ψ ( f A ) is a fuzzy soft r -open. Definition 4.6. Let (X, τ 1, E) and (Y, τ 2, F) be two fuzzy soft topological spaces and φ ψ : (X, τ 1, E) (Y, τ 2, F) be a fuzzy soft mapping. If φ ψ is a bijection, fuzzy soft r -continuous and φ ψ is fuzzy soft r -open mapping, then φ ψ is said to be fuzzy soft r -homeomorphism from X to Y. Theorem 4.7. Let (X, τ 1, E) and (Y, τ 2, F) be two fuzzy soft topological spaces and φ ψ : (X, τ 1, E) (Y, τ 2, F) be a bijective and fuzzy soft r-continuous mapping.

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