Intuitionistic Smooth Bitopological Spaces and Continuity

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  • ABSTRACT

    In this paper, we introduce intuitionistic smooth bitopological spaces and the notions of intuitionistic fuzzy semiinterior and semiclosure. Based on these concepts, the characterizations for the intuitionistic fuzzy pairwise semicontinuous mappings are obtained.


  • KEYWORD

    Intuitionistic , Smooth bitopology

  • 1. Introduction and Preliminaries

    Chang [1] introduced the notion of fuzzy topology. Chang’s fuzzy topology is a crisp subfamily of fuzzy sets. However, in his study, Chang did not consider the notion of openness of a fuzzy set, which seems to be a drawback in the process of fuzzification of topological spaces. To overcome this drawback, Šostak [2, 3], based on the idea of degree of openness, introduced a new definition of fuzzy topology as an extension of Chang’s fuzzy topology. This generalization of fuzzy topological spaces was later rephrased as smooth topology by Ramadan [4].

    Çoker and his colleague [5, 6] introduced intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets which were introduced by Atanassov [7]. Mondal and Samanta [8] introduced the concept of an intuitionistic gradation of openness as a generalization of a smooth topology.

    On the other hand, Kandil [9] introduced the concept of fuzzy bitopological spaces as a natural generalization of Chang’s fuzzy topological spaces. Lee and his colleagues [10, 11] introduced the notion of smooth bitopological spaces as a generalization of smooth topological spaces and Kandil’s fuzzy bitopological spaces.

    Lim et al. [12] defined the term “intuitionistic smooth topology,” which is a slight modification of the intuitionistic gradation of openness of Mondal and Samanta, therefore, it is different from ours.

    In this paper, we introduce intuitionistic smooth bitopological spaces and the notions of intuitionistic fuzzy (,)-(r, s)-semiinterior and semiclosure. Based on these concepts, the characterizations for the intuitionistic fuzzy pairwise (r, s)-semicontinuous mappings are obtained.

    I denotes the unit interval [0, 1] of the real line and I0 = (0, 1]. A member μ; of IX is called a fuzzy set in X. For any μIX, μc denotes the complement 1-μ. By and we denote constant mappings on X with value of 0 and 1, respectively.

    Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair A=(μA, γA) where the functions μA : XI and γA : XI denote the degree of membership and the degree of nonmembership, respectively, and μA + γA ≤ 1. Obviously, every fuzzy set μ in X is an intuitionistic fuzzy set of the form (μ,-μ). I(X) denotes a family of all intuitionistic fuzzy sets in X and “IF” stands for intuitionistic fuzzy.

    Definition 1.1. ( [4]) A smooth topology on X is a mapping T : IXI which satisfies the following properties: (1) (2) (3) The pair (X, T) is called a smooth topological space.

    Definition 1.2. ( [11]) A system (X, T1, T2) consisting of a set X with two smooth topologies T1 and T2 on X is called a smooth bitopological space.

    Definition 1.3. ( [5]) An intuitionistic fuzzy topology on X is a family T of intuitionistic fuzzy sets in X which satisfies the following properties: (1) (2) If , then (3) If for each i, then The pair (X, T) is called an intuitionistic fuzzy topological space.

    2. Intuitionistic Smooth Bitopological Spaces

    Now, we define the notions of intuitionistic smooth topological spaces and intuitionistic smooth bitopological spaces.

    Definition 2.1. An intuitionistic smooth topology on X is a mapping : I(X)I which satisfies the following properties:

    (1)(0)=(1)=1.(2) (3)

    The pair (X, T ) is called an intuitionistic smooth topological space.

    Let (X, T ) be an intuitionistic smooth topological space. For each , an r-cut

    image

    is an intuitionistic fuzzy topology on X.

    Let (X, T) be an intuitionistic fuzzy topological space and Then the mapping Tr : I(X)I defined by

    image

    becomes an intuitionistic smooth topology on X.

    Definition 2.2. Let A be an intuitionistic fuzzy set in intuitionistic smooth topological space (X, T ) and Then A is said to be

    (1)IF - r -open if (2)IF - r -closed if

    Definition 2.3. Let (X, T ) be an intuitionistic smooth topological space. For and for each AI(X), the IF -r-interior is defined by

    and the IF -r-closure is defined by

    Theorem 2.4. Let A be an intuitionistic fuzzy set in an intuitionistic smooth topological space (X, T ) and Then

    (1)-int(A, r)c=-cl(Ac, r). (2)-cl(A, r)c=-int(Ac, r).

    Proof. It follows from Lemma 2.5 in [13].

    Definition 2.5. A system (X, , ) consisting of a set X with two intuitionistic smooth topologies and on X is called a intuitionistic smooth bitopological space(ISBTS for short). Throughout this paper the indices i, j take the value in {1, 2} and ij.

    Definition 2.6. Let A be an intuitionistic fuzzy set in an ISBTS (X, , ) and r, sI0. Then A is said to be

    (1) an IF (,)-(r, s)-semiopen set if there exist an IF --open set B in X such that B ⊆ A⊆ -cl(B, s), (2) an IF (,)-(r, s)-semiopen set if there exist an IF --closed set B in X such that -int(B,s)⊆A⊆B

    Theorem 2.7. Let A be an intuitionistic fuzzy set in an ISBTS (X, , ) and r, sI0. Then the following statements are equivalent:

    (1) A is an IF (, )-(r, s)-semiopen set. (2) Ac is an IF (, )-(r, s)-semiclosed set. (3) -cl(-int(A, r), s) ⊇ A. (4) -int(-cl(Ac, r), s) ⊆ Ac.

    Proof. (1) ⇒ (2) Let A be an (, )-(r, s)-semiopen set. Then there is an IF -r-open set B in X such that BA ⊆ -cl(B, s). Thus -int(Bc, s) ⊆ AcBc. Since Bc is IF -r-closed in X, Ac is a IF (, ) -(r, s)-semiclosed set in X.

    (2) ⇒ (1) Let Ac be an IF (, )-(r, s)-semiclosed set. Then there is an IF -r-closed set B in X such that -int(B, s) ⊆ AcB. Hence BcA ⊆ -cl(Bc, s). Because Bc is IF -r-open in X, A is an IF (, )-(r, s)-semiopen set in X.

    (1) ⇒ (3) Let A be an IF , )-(r, s)-semiopen set in X. Then there exist an IF -r-open set B in X such that BA ⊆ -cl(B, s). Since B is IF -r-open, we have B = -int(B, r) ⊆ -int(A, r). Thus -cl(-int(A, r), s) ⊇ -cl(B, s) ⊇ A.

    (3) ⇒ (1) Let -cl(-int(A, r), s) ⊇ A and take B = -int(A, r). Then B is an IF -r-open set and Hence A is an IF (, )-(r, s)-semiopen set.

    (3) ⇔ (4) It follows from Theorem 2.4.

    Theorem 2.8. Let A be an intuitionistic fuzzy set in an ISBTS (X, , ) and r, sI0. Then (1) If A is IF -r-open in (X, ), then A is an IF (, )- (r, s)-semiopen set in (X,, ). (2) If A is IF -s-open in (X, ), then A is an IF (, )- (s, r)-semiopen set in (X,, ).

    Proof. (1) Let A be an IF -r-open set in (X, ). Then A = -int(A, r). Thus we have -cl(-int(A, r), s) = -cl(A, s) ⊇ A. Hence A is IF (, )-(r, s)-semiopen in (X, , ).

    (2) Similar to (1).

    The following example shows that the converses of the above theorem need not be true.

    Example 2.9. Let X = {x, y} and let A1, A2, A3, and A4 be intuitionistic fuzzy sets in X defined as

    A1(x) = (0.1, 0.7), A1(y) = (0.7, 0.2); A2(x) = (0.6, 0.2), A2(y) = (0.3, 0.6); A3(x) = (0.1, 0.7), A3(y) = (0.9, 0.1);

    and

    A4(x) = (0.7, 0.1), A4(y) = (0.3, 0.6). Define : I(X) → I and : I(X) → I byand Then (, ) is an ISBT on X. Note that and

    Hence A3 is IF (, )-( , )-semiopen and A4 is IF (, )-(, )-semiopen in (X, , ). But A3 is not an IF - -open set in (X, ) and A4 is not an IF --open set in (X, ).

    Theorem 2.10. Let (X,, ) be an ISBTS and r, sI0. Then the following statements are true:

    (1) If {Ak} is a family of IF (, )-(r, s)-semiopen sets in X, then Ak is IF (, )-(r, s)-semiopen.

    (2) If {Ak} is a family of IF (, )-(r, s)-semiclosed sets in X, then Ak is IF (, )-(r, s)-semiclosed.

    Proof. (1) Let {Ak} be a collection of IF (, )-(r, s)-semiopen sets in X. Then for each k,Ak -cl(-int(Ak, r), s).So we haveThus Ak is IF (, )-(r, s)-semiopen.

    (2) It follows from (1) using Theorem 2.7 .

    Definition 2.11. Let (X; , ) be an ISBTS and r, s I0. For each A I(X), the IF (, )-(r, s)-semiinterior is defined by

    and the IF ((, )-(r, s)-semiclosure is defined by

    Obviously, (, )-scl(A, r, s) is the smallest IF (, )-(r, s)-semiclosed set which contains A and (, )-scl(A, r, s)is the greatest IF (, )-(r, s)-semiopen set which is contained in A. Also, (, )-scl(A, r; s) = A for any IF (, )-(r, s)- semiclosed set A and (, )-scl(A, r, s)= A for any IF (, )-(r, s)-semiopen set A.

    Moreover, we have

    image

    Also, we have the following results:

    (1) (, )-scl(, ,r, s) = , (, )-scl(, r, s) = .

    (2) (, )-scl(A, r, s) A.

    (3) (, )-scl(A, r, s)∪ (, )-scl(B, r, s) (, ))-scl(AB, r, s).

    (4) (, )-scl((, )-scl(A, r, s), r, s) = (, )-scl(A, r, s).

    (5) (, )-sint(, r, s) = , (, )-sint(, r, s) = .

    (6) (, )-sint(A, r, s) A.

    (7) (, )-sint(A, r, s) ∩(, )-sint(B, r, s) (, )-sint(AB, r, s).

    (8) (, )-sint((, )-sint(A, r, s), r, s) = (, )-sint(A, r, s).

    Theorem 2.12. Let A be an intuitionistic fuzzy set in an ISBTS (X, , ) and r, s I0. Then we have

    (1) (, )-sint(A, r, s)c = (, )-scl(Ac, r, s).

    (2) (, )-scl(A, r, s)c = (, )-sint(Ac, r, s).

    Proof. (1) Since

    (, ) - sint(A, r, s) A and (, ) - sint(A, r, s)

    is IF (, )-(r, s)-semiopen in X, Ac (, ) - sint(A, r, s)c and (, ) - sint(A, r, s)c is IF (, )-(r, s)-semiclosed. Thus

    image

    From that Ac (, )-scl(Ac, r, s) and (, )-scl(Ac, r, s) is IF (, )-(r, s)-semiclosed, (, )-scl(Ac, r, s)c A and (, )-scl(Ac, r, s)c is IF (, )-(r, s)-semiopen. Thus we have

    image

    Hence

    (, ) - sint(A, r, s)c (, ) - scl(Ac, r, s).

    Therefore

    (, ) - sint(A, r, s)c = (, ) - scl(Ac, r, s).

    (2) Similar to (1).

    3. Continuity in Intuitionistic Smooth Bitopology

    We define the notions of IF pairwise (r, s)-semicontinuous mappings in intuitionistic smooth bitopological spaces, and investigate their characteristic properties.

    Definition 3.1. Let f : (X, ) → (Y, ) be a mapping from an intuitionistic smooth topological spaces X to an intuitionistic smooth topological spaces Y and Then f is called an IF r-continuous mapping if f-1(B) is IF -r-open in X for each IF -r-open set B in Y .

    Definition 3.2. Let f : (X,, ) → (Y, , ) be a mapping from an ISBTS X to an ISBTS Y and r, s I0. Then f is said to be IF pairwise (r, s)-continuous if the induced mapping f : (X, ) → (Y, ) is an IF r-continuous mapping and the induced mapping f : (X, ) → (Y, ) is an IF s-continuous mapping.

    Definition 3.3. Let f : (X,, ) → (Y, , ) be a mapping from an ISBTS X to an ISBTS Y and r, s I0. Then f is said to be IF pairwise (r, s)-semicontinuous if f-1(A) is an IF (, )-(r, s)-semiopen set in X for each IF -r-open set A in Y and f-1(B) is an IF (, )-(s, r)-semiopen set in X for each IF -s-open set B in Y .

    Remark 3.4. It is obvious that every IF pairwise (r, s)-continuous mapping is IF pairwise (r, s)-semicontinuous. But the following example shows that the converse need not be true.

    Example 3.5. Let (X,, ) be an ISBTS as described in Example 2.9. Define : I(X)I and : I(X)I byand

    Then (, ) is an ISBT on X. Consider a mapping f : (X,, ) → (Y, , ) defined by f(x) = x and f(y) = y.

    Then f is IF pairwise (, )-semicontinuous. But f is not an IF pairwise (, )-continuous mapping.

    Theorem 3.6. Let f : (X,, ) → (Y, , ) be a mapping from an ISBTS X to an ISBTS Y and r, s I0. Then the following statements are equivalent:

    (1) f is IF pairwise (r, s)-semicontinuous.

    (2) f-1(A) is an IF (, )-(r, s)-semiclosed set in X for each IF -r-closed set A in Y and f-1(B) is an IF (, )-(s, r)-semiclosed set in X for each IF -s- closed set B in Y .

    (3) For each intuitionistic fuzzy set B in Y ,

    image

    and

    image

    (4) For each intuitionistic fuzzy set A in X,

    f(-int(-cl(A, r), s))-cl(f(A), r)

    and

    f(-int(-cl(A, s), r))-cl(f(A), s).

    Proof. (1) (2) Trivial.

    (2) (3) Let B be an intuitionistic fuzzy set in Y . Then -cl(B, r) is IF -r-closed and -cl(B, s) is IF -s-closed in Y . Hence by (2), f-1(-cl(B, r)) is an IF (, )-(r, s)- semiclosed set and f-1(-cl(B, s)) is an IF (, )-(s, r)- semiclosed set in X. Thus we obtain

    image

    and

    image

    (3) (4) Let A be an intuitionistic fuzzy set in X. Then by (3), we have

    image

    and

    image

    Hence

    f(-int(-cl(A, r), s)) -cl(f(A), r)

    and

    f(-int(-cl(A, s), r)) -cl(f(A), s).

    (4) (2) Let A be any IF -r-closed set and B any IF -s-closed set in Y . By (4), we obtain

    image

    and

    image

    Hence

    image

    and

    image

    Therefore f-1(A) is an IF (, )-(r, s)-semiclosed set and f-1(B) is an IF (, )-(s, r)-semiclosed set in X.

    Theorem 3.7. Let f : (X,, ) → (Y, , ) be a mapping from an ISBTS X to an ISBTS Y and r, s I0. Then the following statements are equivalent:

    (1) f is IF pairwise (r, s)-semicontinuous.

    (2) For each intuitionistic fuzzy set A in X,

    f(, )-scl(A, r, s)) -cl(f(A), r)

    and

    f(, )-scl(A, s, r)) -cl(f(A), s)

    (3) For each intuitionistic fuzzy set B in Y ,

    (, )-scl(f-1(B), r, s) f-1(-cl(B, r))

    and

    (, )-scl(f-1(B), s, r) f-1(-cl(B, s)).

    (4) For each intuitionistic fuzzy set B in Y ,

    f-1(-int(B, r)) (, )-sint(f-1(B), r, s)

    and

    f-1(-int(B, s)) (, )-sint(f-1(B), s, r).

    Proof. (1) (2) Let A be an intuitionistic fuzzy set in X. Then -cl(f(A), r) is IF -r-closed and -cl(f(A), s) is IF -s-closed in Y . Since f is IF pairwise (r, s)-semicontinuous, f-1(-cl(f(A), r)) is an IF (, )-(r, s)-semiclosed set and f-1(-cl(f(A), s)) is an IF (, )-(s, r)-semiclosed set in X. Hence

    image

    and

    image

    Therefore

    f((, )-scl(A, r, s)) -cl(f(A), r)

    and

    f((, )-scl(A, s, r)) -cl(f(A), s)

    (2) (3) Let B be an intuitionistic fuzzy set in Y . Then by (2), we obtain

    image

    and

    image

    Hence

    (, )-scl(f-1(B), r, s) f-1(-cl(B, r))

    and

    (, )-scl(f-1(B), s, r) f-1(-cl(B, s)).

    (3) (4) Let B be an intuitionistic fuzzy set in Y . Then by (3), we have

    (, )-scl(f-1(Bc), r, s) f-1(-cl(Bc, r))

    and

    (, )-scl(f-1(Bc), s, r) f-1(-cl(Bc, s)).

    Hence

    image

    and

    image

    (4) (1) Let A be any IF -r-open set and B any IF -s- open set in Y . Then -int(A, r) = A and -int(B, s) = B. Hence

    image

    and

    image

    Thus

    f-1(A) = (, )-sint(f-1(A), r, s)

    and

    f-1(B) = (, )-sint(f-1(B), s, r).

    Hence f-1(A) is an IF (, )-(r, s)-semiopen set and f-1(B) is an IF (, )-(s, r)-semiopen set in X. Therefore f is IF pairwise (r, s)-semicontinuous.

    Theorem 3.8. Let f : (X,, ) → (Y, , ) be a bijective mapping from an ISBTS X to an ISBTS Y and r, s I0. Then f is IF pairwise (r, s)-semicontinuous if and only if

    image

    and

    image

    for each intuitionistic fuzzy set A in X.

    Proof. Let A be an intuitionistic fuzzy set in X. Since f is one-to-one, by Theorem 3.7, we have

    image

    and

    image

    Because f is onto, we obtain

    image

    and

    image

    Conversely, let B be an intuitionistic fuzzy set in Y . Since f is onto, we obtain

    image

    and

    image

    Because f is one-to-one, we have

    image

    and

    image

    Therefore by Theorem 3.7, f is an intuitionistic fuzzy pairwise (r, s)-semicontinuous mapping.

    Conflict of Interest

    No potential conflict of interest relevant to this article was reported.

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