Intuitionistic Smooth Bitopological Spaces and Continuity
 Author: Kim Jin Tae, Lee Seok Jong
 Organization: Kim Jin Tae; Lee Seok Jong
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 14, Issue1, p49~56, 25 March 2014

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 andI _{0} = (0, 1]. A memberμ ; ofI ^{X} is called afuzzy set in X. For anyμ ∈I ^{X},μ ^{c} denotes the complement 1μ . By and we denote constant mappings on X with value of 0 and 1, respectively.Let
X be a nonempty set. Anintuitionistic fuzzy set A is an ordered pair A=(μA, γA) where the functionsμ _{A} :X →I andγ _{A} :X →I denote the degree of membership and the degree of nonmembership, respectively, andμ _{A} +γ _{A} ≤ 1. Obviously, every fuzzy setμ inX is an intuitionistic fuzzy set of the form (μ ,μ ).I(X) denotes a family of all intuitionistic fuzzy sets inX and “IF” stands for intuitionistic fuzzy.Definition 1.1. ( [4]) Asmooth topology onX is a mappingT :I^{X} →I which satisfies the following properties: (1) (2) (3) The pair( is calledX, T )a smooth topological space. Definition 1.2. ( [11]) A system (X ,T _{1},T _{2}) consisting of a setX with two smooth topologiesT _{1} andT _{2} onX is called asmooth bitopological space. Definition 1.3. ( [5]) Anintuitionistic fuzzy topology onX is a familyT of intuitionistic fuzzy sets inX which satisfies the following properties: (1) (2) If , then (3) If for each i, then The pair (X, T ) is called anintuitionistic 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. Anintuitionistic smooth topology onX is a mapping :I(X) →I which satisfies the following properties:(1)(0)=(1)=1.(2) (3)
The pair (
X, T ) is called anintuitionistic smooth topological space .Let (
X, T ) be an intuitionistic smooth topological space. For each , anr cutis an intuitionistic fuzzy topology on
X .Let (
X, T ) be an intuitionistic fuzzy topological space and Then the mappingT ^{r} :I(X) →I defined bybecomes an intuitionistic smooth topology on
X .Definition 2.2. LetA be an intuitionistic fuzzy set in intuitionistic smooth topological space(X, T ) and ThenA 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 eachA ∈I(X) , theIF rinterior is defined byand the
IF rclosure is defined byTheorem 2.4. LetA 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 setX with two intuitionistic smooth topologies and onX is called aintuitionistic smooth bitopological space (ISBTS for short). Throughout this paper the indicesi, j take the value in {1, 2} andi ≠j .Definition 2.6. LetA be an intuitionistic fuzzy set in an ISBTS (X , , ) andr, s ∈I _{0}. 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. LetA be an intuitionistic fuzzy set in an ISBTS (X , , ) andr, s ∈I _{0}. 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) LetA be an (, )(r, s )semiopen set. Then there is an IF r open setB inX such thatB ⊆A ⊆ cl(B, s ). Thus int(B^{c}, s ) ⊆A^{c} ⊆B^{c} . SinceB^{c} is IF r closed inX ,A^{c} is a IF (, ) (r, s )semiclosed set inX. (2) ⇒ (1) Let
A^{c} be an IF (, )(r, s )semiclosed set. Then there is an IF r closed setB inX such that int(B, s ) ⊆A^{c} ⊆B . HenceB^{c} ⊆A ⊆ cl(B^{c}, s ). BecauseB^{c} is IF r open inX ,A is an IF (, )(r, s )semiopen set inX .(1) ⇒ (3) Let
A be an IF , )(r, s )semiopen set inX . Then there exist an IF r open setB inX such thatB ⊆A ⊆ cl(B, s ). SinceB is IF r open, we haveB = 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 takeB = int(A, r ). ThenB is an IF r open set and HenceA is an IF (, )(r, s )semiopen set.(3) ⇔ (4) It follows from Theorem 2.4.
Theorem 2.8. LetA be an intuitionistic fuzzy set in an ISBTS (X , , ) andr, s ∈I _{0}. Then (1) If A is IF ropen in (X, ), then A is an IF (, ) (r, s)semiopen set in (X,, ). (2) If A is IF sopen in (X, ), then A is an IF (, ) (s, r)semiopen set in (X,, ).Proof. (1) LetA be an IF r open set in (X , ). ThenA = int(A, r ). Thus we have cl(int(A, r), s) = cl(A, s) ⊇ A. HenceA 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. LetX = {x, y } and letA _{1},A _{2},A _{3}, andA _{4} be intuitionistic fuzzy sets inX defined asA1(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 onX . Note that andHence
A _{3} is IF (, )( , )semiopen andA _{4} is IF (, )(, )semiopen in (X , , ). ButA _{3} is not an IF  open set in (X , ) andA _{4} is not an IF open set in (X , ).Theorem 2.10. Let (X ,, ) be an ISBTS andr, s ∈I _{0}. Then the following statements are true:(1) If {
A_{k} } is a family of IF (, )(r, s )semiopen sets inX , thenA_{k} is IF (, )(r, s )semiopen.(2) If {
A_{k} } is a family of IF (, )(r, s )semiclosed sets inX , thenA_{k} is IF (, )(r, s )semiclosed.Proof. (1) Let {A_{k} } be a collection of IF (, )(r, s )semiopen sets inX . Then for eachk ,Ak cl(int(Ak, r), s).So we haveThusA_{k} is IF (, )(r, s )semiopen.(2) It follows from (1) using Theorem 2.7 .
Definition 2.11. Let (X ; , ) be an ISBTS andr, s I _{0}. For eachA I (X ), theIF (, )(r, s )semiinterior is defined byand the
IF ((, )(r, s )semiclosure is defined byObviously, (, )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 setA and (, )scl(A, r, s )=A for any IF (, )(r, s )semiopen setA .Moreover, we have
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(A ∪B, 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(A ∩B, r, s ).(8) (, )sint((, )sint(
A, r, s), r, s ) = (, )sint(A, r, s ).Theorem 2.12. LetA be an intuitionistic fuzzy set in an ISBTS (X , , ) andr, s I _{0}. Then we have(1) (, )sint(
A, r, s )^{c} = (, )scl(A^{c}, r, s ).(2) (, )scl(
A, r, s )^{c} = (, )sint(A^{c}, r, s ).Proof. (1) Since(, )  sint(
A, r, s )A and (, )  sint(A, r, s )is IF (, )(
r, s )semiopen inX ,A^{c} (, )  sint(A, r, s )^{c} and (, )  sint(A, r, s )^{c} is IF (, )(r, s )semiclosed. ThusFrom that
A^{c} (, )scl(A^{c}, r, s ) and (, )scl(A^{c}, r, s ) is IF (, )(r, s )semiclosed, (, )scl(A^{c}, r, s )^{c} A and (, )scl(A^{c}, r, s )^{c} is IF (, )(r, s )semiopen. Thus we haveHence
(, )  sint(
A, r, s )^{c} (, )  scl(A^{c}, r, s ).Therefore
(, )  sint(
A, r, s )^{c} = (, )  scl(A^{c}, 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. Letf : (X , ) → (Y , ) be a mapping from an intuitionistic smooth topological spacesX to an intuitionistic smooth topological spacesY and Thenf is called anIF rcontinuous mapping iff ^{1}(B ) is IF r open inX for each IF r open setB inY .Definition 3.2. Letf : (X ,, ) → (Y , , ) be a mapping from an ISBTSX to an ISBTSY andr, s I _{0}. Thenf is said to beIF pairwise ( if the induced mappingr, s )continuousf : (X , ) → (Y , ) is an IFr continuous mapping and the induced mappingf : (X , ) → (Y , ) is an IFs continuous mapping.Definition 3.3. Letf : (X ,, ) → (Y , , ) be a mapping from an ISBTSX to an ISBTSY andr, s I _{0}. Thenf is said to beIF pairwise (r, s)semicontinuous iff ^{1}(A ) is an IF (, )(r, s )semiopen set inX for each IF r open setA inY andf ^{1}(B ) is an IF (, )(s, r )semiopen set inX for each IF s open setB inY .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 byandThen (, ) is an ISBT on
X . Consider a mappingf : (X ,, ) → (Y , , ) defined byf(x) =x andf(y) =y .Then
f is IF pairwise (, )semicontinuous. Butf is not an IF pairwise (, )continuous mapping.Theorem 3.6. Letf : (X ,, ) → (Y , , ) be a mapping from an ISBTSX to an ISBTSY andr, s I _{0}. 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 inX for each IF r closed setA inY andf ^{1}(B ) is an IF (, )(s, r )semiclosed set inX for each IF s  closed setB inY .(3) For each intuitionistic fuzzy set
B inY ,and
(4) For each intuitionistic fuzzy set
A inX ,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 inY . Then cl(B, r ) is IF r closed and cl(B, s ) is IF s closed inY . Hence by (2),f ^{1}(cl(B, r )) is an IF (, )(r, s ) semiclosed set andf ^{1}(cl(B, s )) is an IF (, )(s, r ) semiclosed set inX . Thus we obtainand
(3) (4) Let
A be an intuitionistic fuzzy set inX . Then by (3), we haveand
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 andB any IF s closed set inY . By (4), we obtainand
Hence
and
Therefore
f ^{1}(A ) is an IF (, )(r ,s )semiclosed set andf ^{1}(B ) is an IF (, )(s ,r )semiclosed set inX .Theorem 3.7. Letf : (X ,, ) → (Y , , ) be a mapping from an ISBTSX to an ISBTSY andr, s I _{0}. Then the following statements are equivalent:(1)
f is IF pairwise (r, s )semicontinuous.(2) For each intuitionistic fuzzy set
A inX ,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 inY ,(, )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 inY ,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) LetA be an intuitionistic fuzzy set inX . Then cl(f (A ),r ) is IF r closed and cl(f (A ),s ) is IF s closed inY . Sincef is IF pairwise (r, s )semicontinuous,f ^{1}(cl(f (A ),r )) is an IF (, )(r, s )semiclosed set andf ^{1}(cl(f (A ),s )) is an IF (, )(s, r )semiclosed set inX . Henceand
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 inY . Then by (2), we obtainand
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 inY . Then by (3), we have(, )scl(
f ^{1}(B^{c} ),r ,s )f ^{1}(cl(B^{c} ,r ))and
(, )scl(
f ^{1}(B^{c} ),s ,r )f ^{1}(cl(B^{c} ,s )).Hence
and
(4) (1) Let
A be any IF r open set andB any IF s  open set inY . Then int(A, r ) =A and int(B, s ) =B . Henceand
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 andf ^{1}(B ) is an IF (, )(s ,r )semiopen set inX . Thereforef is IF pairwise (r, s )semicontinuous.Theorem 3.8. Letf : (X ,, ) → (Y , , ) be a bijective mapping from an ISBTSX to an ISBTSY andr ,s I _{0}. Thenf is IF pairwise (r, s )semicontinuous if and only ifand
for each intuitionistic fuzzy set
A inX .Proof. LetA be an intuitionistic fuzzy set inX . Sincef is onetoone, by Theorem 3.7, we haveand
Because
f is onto, we obtainand
Conversely, let
B be an intuitionistic fuzzy set inY . Sincef is onto, we obtainand
Because
f is onetoone, we haveand
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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