Cmjv01i06p0702

J. Comp. & Math. Sci. Vol. 1 (6), 702-709 (2010)

Fuzzy Boundary Closed Sets and Continuous Functions in Fuzzy Bitopological Space S.S. BENCHALLI1, P.G. PATIL2, and T.D. RAYANAGOUDAR3 1

2

Department of Mathematics, Karnataka University Dharwad-03, Karnataka, India Department of Mathematics,SKSVM Agadi College of Engineering & Technology, Laxmeshwar- 582116, Karnataka, India.

3 Department of Mathematics, Government First Grade College Annigeri-582 201, Karnataka, India.

Email: benchalli_maths@yahoo.com, pgpatil01@gmail.com, rgoudar1980@gmail.com.

ABSTRACT In this paper we introduce a new class of closed sets called fuzzy boundary closed sets in fuzzy bitopological s p ac e s and studied s o m e of their p r o p e r t ie s . Also we introduce two new spaces namely, fuzzy (τi , τj )-b-spaces and fuzzy (τi , τj )bT-spaces as an application. Finally we define the notions of fuzzy boundary continuous and fuzzy strongly boundary bi-continuous maps in fts and some of their properties have been investigated.

2000 Mathematics Subject Classification: 54C08, 54A40.

Key words and phrases: fuzzy (τi , τj )-b-closed set, fuzzy (τi , τj )-b-open set,(τi , τj )b-spaces (τi , τj )-bT- spaces,(τi , τj ) − σk -b-continuous, fuzzy strongly bi-continuous.

1. INTRODUCTION The concept of fuzzy set and fuzzy sets operations were first introduced by paper10.

Zadeh in his classical Balasubramanian and Sundaram1 introduced and studied fuzzy generalized closed sets in fuzzy topological spaces.

Kandil5 introduced and studied the notion of fuzzy bitopological spaces as a natural generalization of bitopological spaces.

Sundaram and Pushpalatha8 introduced fuzzy generalized closed sets and their maps in fuzzy bitopological spaces.

Recently Benchalli4 introduced the concept of fuzzy boundary closed sets and fuzzy boundary continuous maps in fuzzy topological spaces in the year 2002. In this paper we introduce the notion of fuzzy

boundary closed sets in fuzzy bitopological spaces. Also we introduce two new spaces namely, fuzzy (τi , τj )-b-spaces and fuzzy (τi, τj )-bT-spaces as an application. Finally we define the notions of fuzzy boundary continuous and fuzzy strongly boundary bi-continuous maps in fts and some of their properties have been investigated.

2. PRELIMINARIES In the study of a fuzzy bitopological spaces (X, τi , τj ) where i, j ∈ {1, 2} denotes a fuzzy bitopological spaces. We denote the closure and interior of a fuzzy set A with respect to the fuzzy topologies τi , τj in a fuzzy bitopological spaces (τi , τj) by τi − cl(A) and τi − int(A).

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Before entering into our work we recall the following definitions, which are due to various authors.

Definition 2..1. A fuzzy set A of a fts (X, T) is called: 1. a fuzzy generalized closed (g - closed) set1 if cl(A) ≤ U whenever A ≤ U and U is fuzzy open set in (X, τ ).

2. a fuzzy boundary closed (f b - closed) set4 if bd(A) ≤ U whenever A ≤ U and U is fuzzy open set in (X, τ).

The compliment of fuzzy g-closed (resp.f b - closed) set is called fuzzy gopen (resp.f b - open) set.

Definition 2.2. Let i, j ∈ {1, 2} be fixed integers. In a bitopological space (X, τi , τj ) a subset A of (X, τ1 , τ2 ) is said to be (τi , τj )-g-closed2 if τj − cl(A) ⊆ U whenever A ⊆ U and U ∈ τi .

Definition 2..3. Let (X, τ1 , τ2 ) be a fuzzy bitopological space. A fuzzy set A in a X is called fuzzy (τi , τj )-generalized closed9 set if τj − cl(A) ≤ U whenever A ≤ U and U is τi -fuzzy open set.

Definition 2.4. Let i, j ∈ {1, 2} be fixed integers. A bitopological space (X, τ1, τ2 ) is said to be (τi , τj )-τ1/2 -space2 if any (τi, τj )-g- closed set is τj -closed.

Definition 2.5. A map f: (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is called fuzzy τj − σk continuous8 if the inverse image of every σk -fuzzy closed set in (Y, σ1 , σ2 ) is a fuzzy τj closed set in (X, τ1 , τ2 ).

Definition 2.6. A map f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is called fuzzy (τi , τj ) − σk -gcontinuous9 if the inverse image of every σk -fuzzy closed set in (Y, σ1 , σ2 ) is a fuzzy (τi , τj )-g-closed set in (X, τ1 , τ2 ).

Definition 2.7. A map f: (X, τ1 , τ2 )

→ (Y, σ1 , σ2 ) is called fuzzy pair wise continuous 9 if f is fuzzy τ1 − σ1 -continuous and fuzzy τ2 − σ2 -continuous.

Definition 2.8. A map f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is called fuzzy strongly bicontinuous9 if f is fuzzy pair wise continuous, fuzzy τ1 − σ2 -continuous and fuzzy τ2 − σ1 -continuous.

3 . FUZZY (τi , τj )-BOUNDARY CLOSED SETS In this section we introduce fuzzy (τi , τj)-boundary closed sets in fuzzy bitopological space and study some of their properties.

Definition 3.1. Let i, j ∈ {1, 2} be fixed integers. In a fuzzy bitopological space (X, τ1 , τ2 ), a fuzzy set A of X is said to be fuzzy (τi , τj )-boundary closed (briefly fuzzy (τi , τj )-b-closed) set if τj − bd (A) ≤ U whenever A ≤ U and U is τi -fuzzy open set.

We denote the family of all fuzzy (τi , τj )-b-closed sets in fuzzy bitopological spaces by β(τi , τj ) Remark 3.2. By setting τ1 = τ2 in the definition 3.1, fuzzy (τi , τj )-boundary closed set is b-closed fuzzy set in the sense of 4.

Theorem 3.3. If A is τj -closed fuzzy set of (X, τ1 , τ2 ), then it is fuzzy (τi , τj )-bclosed set.

Proof. Let G be a τi -open fuzzy set such that A ≤ G. Since A is τj -closed fuzzy set and thus τj − cl(A) = A ≤ G. But τ j − bd(A) ≤ τj− cl(A). Therefore τ j − bd(A) ≤ G. Hence A is fuzzy (τi , τj )-b-closed set.

The converse of the above theorem need not be true as seen from the

Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)

S. S. Benchalli et al., J. Comp. & Math. Sci. Vol. 1 (6), 702-709 (2010)

following example.

Example 3.4. Let X = {a, b, c}, fuzzy sets A, B and C be defined as follows: A ={(a, 1), (b, 1), (c, 0)}, B = {(a, 1), (b, 0), (c, 0)}and C = {(a, 0), (b, 1), (c, 0)}.

Consider τ 1 = {0, 1, A} and τ2 = {0, 1, B}. Then (X, τ1 , τ2 ) is fuzzy bitopological space. Then the set C is fuzzy (τ2 , τ1 )-bclosed set. But C is not τ1 -closed fuzzy set.

Theorem 3.5. If A and B are fuzzy (τi , τj)-b-closed sets, then A ∨ B is also fuzzy (τi, τj )- b-closed set.

Remark 3.6. Finite union of fuzzy (τi , τj )b-closed sets is fuzzy (τi , τj )-b-closed set.

Remark 3.7. Intersection o f two fuzzy (τi , τj )-b-closed sets need not be a fuzzy (τi , τj )-b- closed set as seen from the following example.

Example 3.8. Let X = {a, b, c}, fuzzy sets A, B and C be defined as follows: A ={(a, 1), (b, 1), (c, 0)}, B = {(a, 1), (b, 0), (c, 0)} and C = {(a, 1), (b, 0), (c, 1)}.

Consider τ1 = {0, 1, A} and τ2 = {0, 1, C}.Then (X, τ1 , τ2 ) is a fuzzy bitopological space. Then the sets A and C are fuzzy (τ2 , τ1 )-b-closed sets, but A ∧ C = B is not a fuzzy (τ2 , τ1 )-b-closed set.

Remark 3.9. Fuzzy (τ1 , τ2 )-b-closed sets are generally not equal to fuzzy (τ2 , τ1 )-bclosed sets as seen from the following example.

Example 3.10. Let X = {a, b, c}, fuzzy sets A, B, C , D and E are defined as follows: A = {(a, 1), (b, 0), (c, 0)}, B = {(a, 0), (b, 1), (c, 1)}, C = {(a, 0), (b, 1), (c, 0)}, D = {(a, 0), (b, 0), (c, 1)}and C

704

={(a, 1), (b, 0), (c, 1)}.Consider τ1 ={0, 1, B, C, D} and τ2 = {0, 1, A, B}.Then (X, τ1 , τ2 ) is a fuzzy bitopological space.

Then the set C is fuzzy (τ2 , τ1 )-b-closed set, but not fuzzy (τ1 , τ2 )-b-closed set.

Theorem 3.11. If τ1 ≤ τ2 in (X, τ1 , τ2 ), then β(τ1 , τ2 ) ≥ β(τ2 , τ1 ) where β(τi, τj ) denotes the class of all fuzzy (τi , τj )-bclosed set.

Proof. Let A be a fuzzy (τ2 , τ1 )-b-closed set and G be a τ1 -fuzzy open set such that A ≤ G. Since τ1 ≤ τ2 , it follows that G is τ2 - fuzzy open set and τ2 − bd(A) ≤ τ1 − bd(A). Then τ2 − bd(A) ≤ G. Therefore A is fuzzy (τ2 , τ1 )-b-closed set. Hence β(τ1, τ2 ) ≥ β(τ2 , τ1 ).

Theorem 3.12. If A is fuzzy (τi , τj )-bclosed set and B is τj - fuzzy closed set, then A ∨ B is also fuzzy (τi , τj )-b-closed set.

Proof. The proof is follows from the Theorem 3.3 and Theorem 3.5.

Theorem 3.13. Every fuzzy (τi , τj )-gclosed set is fuzzy (τi , τj )-b-closed set.

Proof. Let A be a fuzzy (τi , τj )-g-closed set in (X, τ1 , τ2 ). Let G be a τi - fuzzy open set in (X, τ1 , τ2 ) such that A ≤ G. Since A is fuzzy (τi , τj )-g-closed, τj − cl(A) ≤ G. But τj − bd(A) ≤ τj − cl(A).Therefore τj − bd(A) ≤ G. Hence A is fuzzy (τi , τj )-bclosed set.

The converse of the above theorem is true if τj − cl(A) ∧ τj − cl(1 − A) > 0 for every fuzzy set A. Theorem 3.14. If A is fuzzy (τi , τj )-bclosed set and τj − cl(A) ∧ τj − cl(1 − A) > 0, then A is fuzzy (τi , τj )-g-closed set.

Proof. Let G be a τi - fuzzy open set such that A ≤ G. Since A is fuzzy (τi , τj )-b-

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closed set, then τj − bd(A) ≤ G. Also since τj − cl(A) ∧ τj − cl(1 − A) > 0, then τj − bd(A) = τj − cl(A) [10]. Therefore τj − cl(A) ≤ G which implies A is fuzzy (τi , τj)g-closed set.

Theorem 3.15. If A is τi - fuzzy open set and fuzzy (τi , τj )-b-closed in (X, τ1 , τ2 ), then A is τj - fuzzy closed set in (X, τ1 , τ2 ).

Proof. Let A be a fuzzy set in (X, τ1 , τ2 ) which is τi - fuzzy open set and fuzzy (τi , τj )-b- closed. We have A ≤ A. Then τj − bd(A) ≤ A. Then by [10 Theorem 3.6], A is τj - fuzzy closed set in (X, τ1 , τ2 ).

Theorem 3.16. If A is fuzzy set in (X, τ1 , τ2 ) is both τi - fuzzy open set and fuzzy (τi , τj )-b- closed set, then A is fuzzy (τi , τj )-gclosed set.

Proof. Since A is τi - fuzzy open set and fuzzy (τi , τj )-b-closed set. Then by Theorem 3.15, A is τj - fuzzy closed set in (X, τ1 , τ2 ).

So A is fuzzy (τi , τj )-g-closed set.

Theorem 3.17. If A is fuzzy set in (X, τ1, τ2 ) such that τj − bd(A) ∧ (1 − τj − bd(A)) = 0. Then A is fuzzy (τi , τj )-b-closed set if and only if τj − bd(A) ∧ (1 − A) contains no non-zero τi - fuzzy closed sets.

Proof. Let A be any τi - fuzzy closed set such that F ≤ τj − bd(A) ∧ (1 − A). Now F ≤ 1 − A which implies A ≤ 1 − f, 1 − F is τi fuzzy open set. Since A is fuzzy (τi , τj )b-closed we have τj − bd(A) ≤ 1 − F which implies that F ≤ 1 − τj − bd(A).Thus F ≤ τj − bd(A) and F ≤ (1 − τj − bd(A)).

Therefore by hypothesis, F ≤ 1 − τj − bd(A) ∧ (1 − τj − bd (A)) = 0. Therefore F = 0.

Conversely, let A ≤ U where U is τi - fuzzy open set. If τj −bd(A) ≥ U then τj −bd(A) ∧(1 −U ) is τi - fuzzy closed set and

τj−bd(A)∧(1−U ) ≤ τj −bd(A)∧(1−A) which contradicts to hypothesis. Hence A is fuzzy (τi , τj )-b-closed set.

Definition 3.18. A fuzzy set A in a fuzzy bitopological s p a c e (X, τ1 , τ2 ) is called fuzzy (τi , τj )-b-open set if its complement (1 − A) is fuzzy (τi , τj )-bclosed set.

Theorem 3.19. Every τj - fuzzy open set is fuzzy (τi , τj )-b-open.

Proof. The proof is follows from the Theorem 3.3.

Example 3.20. In Example 3.4, the fuzzy set C is fuzzy (τ2 , τ1 )-b-open set but not τ1 – fuzz set open.

Theorem 3.21. If A and B are fuzzy (τi , τj )-b-open sets in (X, τ1 , τ2 ), then A ∧ B is fuzzy open set.

Proof. Follows from the Theorem 3.5.

4. APPLICATIONS OF (τi , τj )-bCLOSED SET In this section we define and study the new spaces namely, fuzzy (τi , τj )-bspaces and fuzzy (τi , τj )-bT-spaces as an application of fuzzy (τi , τj )-b-closed set in bitopological spaces.

Definition 4.1. A fuzzy bitopological space (X, τ1 , τ2 ) is said to be fuzzy (τi , τj )b-space if every fuzzy (τi , τj )-b-closed set is τj - fuzzy closed set.

Theorem 4.2. A fuzzy bitopological space (X, τ1 , τ2 ) is fuzzy (τi , τj )-b-space if and only if every fuzzy (τi , τj )-b-open set is τj open set.

Theorem 4.3. Every fuzzy (τi , τj )-b-space is fuzzy (τi , τj )-T1/2 -space.

Definition 4.4. A fuzzy bitopological space (X, τ1 , τ2 ) is said to be fuzzy (τi , τj)bT-space if every fuzzy (τi , τj )-b-closed set

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is fuzzy (τi , τj )-g-closed set.

Theorem 4.5. Every fuzzy (τi , τj )-b-space is fuzzy (τi , τj )-bT-space.

The converse of the above theorem need not be true as seen from the following example.

Example 4.6. Let X = {a, b, c}, Fuzzy sets A, B and C be defined as follows: A = (a, 1), (b, 0), (c, 0)}, B = {(a, 0), (b, 1), (c, 0)} and C = {(a, 0), (b, 1), (c, 1)}.

Consider τ1 = {0, 1, A, C } and τ2 = {0, 1, A}. Then (X, τ1 , τ2 ) is a fuzzy bitopological space. Then (X, τ1 , τ2 ) is fuzzy (τ1 , τ2 )-bTspace but not fuzzy (τ1 , τ2)-b-space as the fuzzy set B is fuzzy (τ1 , τ2 )-b-closed set but not τ2 -closed fuzzy set.

Theorem 4.7. A fuzzy bitopological space (X, τ1 , τ2 ) is fuzzy (τi , τj )-b-space if and only if it is fuzzy (τi , τj )-T1/2 -space and fuzzy (τi , τj )-bT-space.

Proof. Suppose X is fuzzy (τi , τj )-bspace. Then by Theorems 4.11 and 4.13, (X, τ1 , τ2 ) is fuzzy (τi , τj )-T1/2 -space and fuzzy (X, τ1 , τ2 ) is fuzzy (τi , τj )-bT-space.

Conversely, let G fuzzy (τi , τj )-b-closed.

Since X is fuzzy (τi , τj )-bT-space, G is fuzzy (τi , τj )-g-closed set. Again since X is fuzzy (τi , τj )-T1/2 -space, G is τj-closed fuzzy set. Hence X is (τi , τj )-b-space.

5. FUZZY BOUNDARY CONTINUOUS Maps In this section, we introduce fuzzy boundary-continuous and fuzzy boundary pair wise-irresolute maps in fuzzy bitopological spaces and obtain some of their properties.

Definition 5.1. A map f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is called fuzzy (τi , τj ) − σk -b-

continuous if the inverse image of every σk -fuzzy closed set in (Y, σ1 , σ2 ) is a fuzzy (τi , τj )-b-closed set in (X, τ1 , τ2 ).

Remark 5.2. Suppose that τ1 = τ2 = τ and σ1 =σ2 =σ, in Definition 5.1, then fuzzy (τi , τj )−σk -b-continuous map coincide with the fuzzy b-continuous in fuzzy topological space due to [4].

Theorem 5.3. If a map f: (X, τ1 , τ2 ) → (Y σ1 , σ2 ) is fuzzy τj , σk -continuous map then f is fuzzy (τi , τj ) − σk -b-continuous.

Proof. Let G be σk - closed fuzzy set Y. −1

Since f is fuzzy τj −σk –continuous f (G) is fuzzy τj -closed set in X . Since every τj fuzzy closed set is fuzzy (τi , τj)-b-closed −1

set, f (G) is fuzzy (τi , τj )-b-closed set in X . Hence f is fuzzy (τi , τj ) − σk -bcontinuous.

The converse of the above theorem need not be true as seen from the following example.

Example 5.4. Let X = Y {a, b, c}.

Fuzzy subsets A, B, C , D and E be defined as follows: A = {(a, 1), (b, 0), (c, 0)}, B = (a, 1), (b, 1), (c, 0)}, C = {(a, 1), (b, 0), (c, 1)}, D = {(a, 0), (b, 1), (c, 0)} and E ={(a, 0), (b, 1), (c, 1)}.Consider τ1 = {0, 1, A}, τ2 = {0, 1, B}, σ1 = {0, 1, C } and σ2 = {0, 1, D, E}. Therefore (X, τ1 , τ2 ) and (Y, σ1 , σ2 ) are fuzzy bitopological spaces. Let f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) be the identity map. Then f is fuzzy (τ1 , τ2 ) − σ1 -b- continuous but not fuzzy τ2 − σ1 continuous.

Theorem 5.5. If a map f: (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is fuzzy (τi , τj ) − σk -bcontinuous and (X, τ1 , τ2) is fuzzy (τi , τj )-bspace, then f is fuzzy τj − σk -continuous.

Proof. Let G be fuzzy σk -closed set in (Y,

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707

σ1 , σ2 ). Since f is fuzzy (τi , τj )−σk -b-

σ1 , σ2 ). Since f is fuzzy (τi , τj )−σk -g-

continuous, f (G) is fuzzy (τi , τj )-bclosed set in (X, τ1 , τ2 ). Again since (X, τ1 ,

continuous, f (G) is fuzzy (τi , τj )-g-closed

−1

−1

τ2 ) is (τi , τj )-b-space, f (G) is fuzzy τj closed set in (X, τ1 , τ2 ). Hence f is fuzzy τj − σk -continuous.

Theorem 5.6. A map f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is fuzzy (τi , τj ) − σk -b-continuous if and only if the inverse image of every fuzzy σk -closed set in (Y, σ1 , σ2 ) is fuzzy (τi, τj )-b-open set in (X, τ1 , τ2 ).

Proof. Suppose f: (X, τ1 , τ2 ) → (Y, σ1 , σ2) is fuzzy (τi , τj ) − σk -b-continuous. Let G be fuzzy σk -open set in (Y σ1 , σ2 ),1 − G is σk - fuzzy closed set in (Y, σ1 , σ2 ). Since −1

f is fuzzy (τi , τj ) − σk -b-continuous, f (1 − G) is fuzzy (τi , τj )-b-closed set in (X, τ1 , τ2 ).

So f

−1

(1 − G) = 1 − f

−1

(G) is fuzzy (τi , τj )−1

b-closed set in (X, τ1 , τ2 ). Therefore f (G) is fuzzy (τi , τj )-b-open set in (X, τ1 , τ2 ).

Conversely, if the inverse image of every σk fuzzy open set is fuzzy (τi , τj )-b-open set.

Let G be the σk - fuzzy closed set in (Y, σ1 , σ2 ), 1 − G is fuzzy σk -open set in (Y, σ1 , σ2 ).

By hypothesis, f

−1

(1 − G) is fuzzy (τi , τj )-b−1

open set in (X, τ1 , τ2 ). So f (1 − G) = 1 − −1

f (G) is fuzzy (τi , τj )- b-open set in (X, τ1 , τ ). Therefore f −1 (G) is fuzzy (τ , τ )-b2

i

j

closed set in (X, τ1 , τ2 ). Hence f is fuzzy (τi , τj ) − σk -b-continuous.

Theorem 5.7. If a map f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is fuzzy (τi , τj ) − σk -gcontinuous, then f is f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is fuzzy (τi , τj ) − σk -b-continuous.

Proof. Let G be fuzzy σk -closed set in (Y,

−1

−1

set in (X, τ1 , τ2 ). By Theorem 3.13, f (G) is fuzzy (τi , τj )-b- closed set in (X, τ1 , τ2 ).

Hence f is fuzzy (τi , τj ) − σk -b-continuous.

Theorem 5.8. If a map f: (X, τ1 , τ2 ) → (Y σ1 , σ2 ) is fuzzy (τi , τj ) − σk -b-continuous and (X, τ1 , τ2 ) is fuzzy (τi , τj )-b-space, then f is fuzzy (τi , τj ) − σk -g-continuous.

Proof. Let G be fuzzy σk -closed set in (Y, σ1 , σ2 ). Since f is fuzzy (τi , τj ) − σk -b−1

continuous, f (G) is fuzzy (τi , τj )-g-closed set in (X, τ1 , τ2 ). Again since (X, τ1 , τ2 ) is -1

fuzzy (τi , τj )- b-space, f (G) is fuzzy (τi , τj )-g-closed set in (X, τ1 , τ2). Hence f is fuzzy (τi, τj ) − σk-g- continuous.

Definition 5.9. A map f: (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) is called fuzzy boundary bicontinuous (briefly fuzzy b-bi-continuous) if f is fuzzy (τ1 , τ2 ) − σ2 -b-continuous and fuzzy (τ2 , τ1 ) − σ1 -b- continuous.

Definition 5.10. A map f: (X, τ1 , τ2) → (Y, σ1, σ2 ) is called fuzzy strongly boundary bi- continuous (briefly fuzzy strongly b-bi-continuous) if f is fuzzy b-bicontinuous, fuzzy (τ2 , τ1 ) − σ2 -b-continuous and fuzzy (τ1 , τ2) − σ1 -b-continuous.

Theorem 5.11. If f: (X, τ1 , τ2 ) → (Y, σ1, σ2 ) is pair wise continuous then f is fuzzy b-bi- continuous.

Proof. The proof is follows from the Definitions 2.7 and 5.9.

The converse of the above theorem need not be true as seen from the following example.

Example 5.12. Let X = Y = {a, b, c}.

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Fuzzy subsets A, B, C and D be defined as follows: A = {(a, 1), (b, 0), (c, 0)}, B ={(a, 0), (b, 1), (c, 0)}, C = {(a, 1), (b, 1), (c, 0)} and D = {(a, 1), (b, 0), (c, 1)}.

Consider τ1 = {0, 1, A, B, C}, τ2 = {0, 1, A}, σ1 = {0, 1, D} and σ 2 = {0, 1, C}. Therefore ( X, τ1 , τ2) and (Y, σ1 , σ2) are fuzzy bitopological s p a c e s . Let f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) be the identity map.

Then f is fuzzy b-bi-continuous b u t not fuzzy pair wise continuous.

708

defined as follows: A = {(a, 1), (b, 0), (c, 0)}, B = {(p, 1), (q, 0)} and C = {(p, 0), (q, 1)}. Consider τ1 = {0, 1, A}, τ2 = discrete fuzzy topology on X , σ1 = {0, 1, B} and σ2 = {0, 1, C }. Therefore (X, τ1 , τ2) and (Y, σ1 , σ2 ) are fuzzy bitopological spaces. Define a map f: (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) by f (a) = p and f (b) = f (c) = q.Then f is fuzzy (τ1 , τ2) − σ2 -b-continuous a n d f u z z y (τ2, τ1 ) − σ2 -b-continuous but not strongly fuzzy (τ2, τ1 ) − σ2 -b-continuous.

Therefore f is fuzzy b-bi-continuous but not fuzzy strongly b-bi-continuous.

ACKNOWLEDGEMENT

Theorem 5.13. If f: (X, τ1 , τ2 ) → (Y, σ1 , σ2) is fuzzy strongly bi-continuous then f is fuzzy strongly b-bi-continuous.

Proof. The proof is follows from the Theorem 5.11 and Definitions 2.8 and 5.10.

The converse of the above theorem need not be true as seen from the following example.

Example 5.14. Let X = Y = {a, b, c}.

Fuzzy subsets A, B and C be defined as follows: A = {(a, 1), (b, 0), (c, 0)}, B = {(a, 0), (b, 1), (c, 0)} and C = {(a, 1), (b, 1), (c, 0)}. Consider τ1 = {0, 1, A, B, C }, τ2 = {0, 1, A}, σ1 = {0, 1, C } and σ2 = {0, 1, B, C }. Therefore (X, τ1 , τ2 ) and (Y, σ1 , σ2 ) are fuzzy bitopological spaces. Let f: (X, τ1, τ2 ) → (Y, σ1 , σ2 ) be the identity map.

Then f is fuzzy strongly b-bi-continuous but not fuzzy strongly bi-continuous.

Remark 5.15. The following example shows that fuzzy b-bi-continuous m a p need not be a fuzzy strongly b-bi-continuous ma p .

Example 5.16. Let X = {a, b, c}, Y = {p, q}. Fuzzy subsets A, B and C be

The present research work was financially supported by the UGC - SAP1 - DRS - F510 grants of the University Grants Commission, New Delhi, India.

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