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Intuitionistic Smooth Bitopological Spaces and Continuity
Intuitionistic Smooth Bitopological Spaces and Continuity
International Journal of Fuzzy Logic and Intelligent Systems. 2014. Mar, 14(1): 49-56
Copyright © 2014, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : February 26, 2014
  • Accepted : March 19, 2014
  • Published : March 25, 2014
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About the Authors
Jin Tae Kim
Seok Jong Lee

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.
Keywords
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 (
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,
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)-( 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 I 0 = (0, 1]. A member μ ; of I X is called a fuzzy set in X. For any μ I X , μ c denotes the complement 1- μ . By
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and
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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 μ A : X I and γ A : X I denote the degree of membership and the degree of nonmembership, respectively, and μ A + γ A ≤ 1. Obviously, every fuzzy set μ in X is an intuitionistic fuzzy set of the form ( μ ,
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- 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 : IX I 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 , T 1 , T 2 ) consisting of a set X with two smooth topologies T 1 and T 2 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) Iffor eachi, 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
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: 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
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, an r -cut
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is an intuitionistic fuzzy topology on X .
Let ( X, T ) be an intuitionistic fuzzy topological space and
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Then the mapping T r : I(X) I defined by
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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
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Then A is said to be
  • (1)IF- r -openif
  • (2)IF- r -closedif
Definition 2.3. Let (X, T ) be an intuitionistic smooth topological space. For
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and for each A I(X) , the IF
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-r-interior is defined by
and the IF
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- r-closure is defined by
Theorem 2.4. Let A be an intuitionistic fuzzy set in an intuitionistic smooth topological space (X, T ) and
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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 ,
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,
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) consisting of a set X with two intuitionistic smooth topologies
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and
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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 i j .
Definition 2.6. Let A be an intuitionistic fuzzy set in an ISBTS ( X ,
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,
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) and r, s I 0 . Then A is said to be
  • (1) anIF(,)-(r, s)-semiopenset if there exist an IF--open setBinXsuch thatB⊆A⊆-cl(B, s),
  • (2) anIF(,)-(r, s)-semiopenset if there exist an IF--closed setBinXsuch that-int(B,s)⊆A⊆B
Theorem 2.7. Let A be an intuitionistic fuzzy set in an ISBTS ( X ,
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,
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) and r, s I 0 . Then the following statements are equivalent:
  • (1)Ais an IF (,)-(r, s)-semiopen set.
  • (2)Acis 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 (
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,
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)-( r, s )-semiopen set. Then there is an IF
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- r -open set B in X such that B A
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-cl( B, s ). Thus
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-int( Bc, s ) ⊆ Ac Bc . Since Bc is IF
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- r -closed in X , Ac is a IF (
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,
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) -( r, s )-semiclosed set in X.
(2) ⇒ (1) Let Ac be an IF (
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,
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)-( r, s )-semiclosed set. Then there is an IF
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- r -closed set B in X such that
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-int( B, s ) ⊆ Ac B . Hence Bc A
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-cl( Bc, s ). Because Bc is IF
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- r -open in X , A is an IF (
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,
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)-( r, s )-semiopen set in X .
(1) ⇒ (3) Let A be an IF
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,
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)-( r, s )-semiopen set in X . Then there exist an IF
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- r -open set B in X such that B A
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-cl( B, s ). Since B is IF
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- r -open, we have B =
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-int( B, r ) ⊆
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-int( A, r ). Thus
  • -cl(-int(A, r),s) ⊇-cl(B, s) ⊇A.
(3) ⇒ (1) Let
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-cl(
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-int( A, r ), s ) ⊇ A and take B =
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-int( A, r ). Then B is an IF
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- r -open set and
Hence A is an IF (
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,
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)-( 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 ,
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,
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) and r, s I 0 . Then
  • (1) IfAis IF-r-open in (X,), thenAis an IF (,)- (r, s)-semiopen set in (X,,).
  • (2) IfAis IF-s-open in (X,), thenAis an IF (,)- (s, r)-semiopen set in (X,,).
Proof. (1) Let A be an IF
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- r -open set in ( X ,
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). Then A =
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-int( A, r ). Thus we have
  • -cl(-int(A, r),s) =-cl(A, s) ⊇A.
Hence A is IF (
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,
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)-( r, s )-semiopen in ( X ,
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,
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).
(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 A 1 , A 2 , A 3 , and A 4 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
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: I ( X ) → I and
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: I ( X ) → I by
and
Then (
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,
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) is an ISBT on X . Note that
and
Hence A 3 is IF (
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,
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)-(
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,
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)-semiopen and A 4 is IF (
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,
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)-(
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,
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)-semiopen in ( X ,
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,
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). But A 3 is not an IF
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-
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-open set in ( X ,
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) and A 4 is not an IF
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-
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-open set in ( X ,
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).
Theorem 2.10. Let ( X ,
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,
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) be an ISBTS and r, s I 0 . Then the following statements are true:
(1) If { Ak } is a family of IF (
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,
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)-( r, s )-semiopen sets in X , then
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Ak is IF (
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,
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)-( r, s )-semiopen.
(2) If { Ak } is a family of IF (
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,
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)-( r, s )-semiclosed sets in X , then
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Ak is IF (
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,
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)-( r, s )-semiclosed.
Proof. (1) Let { Ak } be a collection of IF (
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,
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)-( r, s )-semiopen sets in X . Then for each k ,
  • Ak-cl(-int(Ak,r),s).
So we have
Thus
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Ak is IF (
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,
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)-( r, s )-semiopen.
(2) It follows from (1) using Theorem 2.7 .
Definition 2.11. Let ( X ;
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,
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) be an ISBTS and r, s
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I 0 . For each A
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I ( X ), the IF (
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,
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)-( r, s )-s emiinterior is defined by
and the IF ((
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,
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)-( r, s )- semiclosure is defined by
Obviously, (
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,
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)-scl( A, r, s ) is the smallest IF (
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,
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)-( r, s )-semiclosed set which contains A and (
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,
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)-scl( A, r, s )is the greatest IF (
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,
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)-( r, s )-semiopen set which is contained in A. Also, (
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,
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)-scl( A, r; s ) = A for any IF (
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,
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)-( r, s )- semiclosed set A and (
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,
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)-scl( A, r, s )= A for any IF (
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,
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)-( r, s )-semiopen set A .
Moreover, we have
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Also, we have the following results:
(1) (
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,
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)-scl(
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, ,r, s ) =
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, (
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,
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)-scl(
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