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Intuitionistic Fuzzy <italic>δ</italic>-continuous Functions
Intuitionistic Fuzzy δ-continuous Functions
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Dec, 13(4): 336-344
Copyright © 2013, 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 : December 02, 2013
  • Accepted : December 23, 2013
  • Published : December 25, 2013
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Yeon Seok Eom
Seok Jong Lee

Abstract
In this paper, we characterize the intuitionistic fuzzy δ -continuous, intuitionistic fuzzy weakly δ -continuous, intuitionistic fuzzy almost continuous, and intuitionistic fuzzy almost strongly θ -continuous functions in terms of intuitionistic fuzzy δ -closure and interior or θ- closure and interior.
Keywords
1. Introduction and Preliminaries
By using the intuitionistic fuzzy sets introduced by Atanassov [1] , Çoker and his colleagues [2 - 4] introduced the intuitionistic fuzzy topological space, which is a generalization of the fuzzy topological space. Moreover, many researchers have studied about this space [5 - 12] .
In the intuitionistic fuzzy topological spaces, Hanafy et al. [13] introduced the concept of intuitionistic fuzzy θ -closure as a generalization of the concept of fuzzy θ -closure by Mukherjee and Sinha [14 , 15] , and characterized some types of functions. In the previous papers [16 , 17] , we also introduced and investigated some properties of the concept of intuitionistic fuzzy θ -interior and δ -closure in intuitionistic fuzzy topological spaces.
In this paper, we characterize the intuitionistic fuzzy δ -continuous, intuitionistic fuzzy weakly δ -continuous, intuitionistic fuzzy almost continuous, and intuitionistic fuzzy almost strongly θ -continuous functions in terms of intuitionistic fuzzy δ -closure and interior, or θ -closure and interior.
Let X be a nonempty set and I the unit interval [0, 1]. An intuitionistic fuzzy set A in X is an object of the form A = ( μA, ϓA ), where the functions μA : X I and ϓA : X I denote the degree of membership and the degree of nonmembership, respectively, and μA + ϓA ≤ 1. Obviously, every fuzzy set μA in X is an intuitionistic fuzzy set of the form ( μA , 1 - μA ).
Throughout this paper, I ( X ) denotes the family of all intuitionistic fuzzy sets in X , and “IF” stands for “intuitionistic fuzzy.” For the notions which are not mentioned in this paper, refer to [17] .
Theorem 1.1 ( [7] ). The following are equivalent:
  • (1) An IF setAis IF semi-open inX.
  • (2)A≤ cl(int(A)).
Corollary 1.2 ( [17] ). If U is an IF regular open set, then U is an IF δ -open set.
Theorem 1.3 ( [17] ). For any IF semi-open set A , we have cl( A ) = cl δ ( A ).
Lemma 1.4 ( [17] ). (1) For any IF set U in an IF topological space ( X, T ), int(cl)undefined( U )) is an IF regular open set.
  • (2) For any IF open setUin an IF topological space (X, T) such thatx(αβ)qU, int(cl(U)) is an IF regular openq-neighborhood ofx(α,β).
Theorem 1.5 ( [12] ). Let x(αβ) be an IF point in X , and U = (μU, ϓU) an IF set in X . Then x (α,β) ∈ cl( U ) if and only if UqN , for any IF q -neighborhood N of x (α,β) .
2. Intuitionistic Fuzzy δ-continuous andWeakly δ-continuous Functions
Recall that a fuzzy set N in ( X, T ) is said to be a fuzzy δ-neighborhood of a fuzzy point xα if there exists a fuzzy regular open q -neighborhood V of xα such that
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or equivalently V N (See [14] ). Now, we define a similar definition in the intuitionistic fuzzy topological spaces.
Definition 2.1. An intuitionistic fuzzy set N in ( X, T ) is said to be an intuitionistic fuzzy δ-neighborhood of an intuitionistic fuzzy point x (α,β) if there exists an intuitionistic fuzzy regular open q -neighborhood V of x (α,β) such that
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Lemma 2.2. An IF set A is an IF δ -open set in ( X, T ) if and only if for any IF point x (α,β) with x(α,β)qA, A is an IF δ -neighborhood of x (α,β) .
Proof . Let A be an IF δ -open set in ( X, T ) such that x(α,β)qA . Then
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Since Ac is an IF δ -closed set, we have x (α,β) Ac = cl δ ( Ac ). Then there exists an IF regular open q -neighborhood U of x (α,β) such that
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Thus U A . Hence A is an IF δ -neighborhood of x (α,β) .
Conversely, to show that Ac is an IF δ -closed set, take any x (α,β) Ac . Then we have x(α,β)qA . Thus A is an IF δ -neighborhood of x (α,β) . Therefore there exists an IF regular open q -neighborhood V of x (α,β) such that V Ac , i.e. x (α,β) ∉ cl δ ( Ac ). Since cl δ ( Ac ) ≤ Ac , we have Ac is an IF δ -closed set. Hence A is an IF δ -open set.
Recall that a function f : ( X, T ) → ( Y, T’ ) is said to be a fuzzy δ-continuous function if for each fuzzy point xα in X and for any fuzzy regular open q -neighborhood V of f(x(α)) , there exists an fuzzy regular open q -neighborhood U of x(α) such that f(U) ≤ V (See [18] ). We define a similar definition in the intuitionistic fuzzy topological spaces as follows.
Definition 2.3. A function f : (X, T) → (Y, T’) is said to be intuitionistic fuzzy δ-continuous if for each intuitionistic fuzzy point x (α,β) in X and for any intuitionistic fuzzy regular open q -neighborhood V of f ( x (α,β) ), there exists an intuitionistic fuzzy regular open q -neighborhood U of x (α,β) such that
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Now, we characterize the intuitionistic fuzzy δ -continuous function in terms of IF δ -closure and IF δ interior.
Theorem 2.4. Let f : (X, T) → (Y, T’) be a function. Then the following statements are equivalent:
  • (1)fis an IFδ-continuous function.
  • (2)f(clδ(U)) ≤ clδ(f(U)) for each IF setUinX.
  • (3) clδ(f-1(V)) ≤f-1(clδ(V)) for each IF setVinY.
  • (4)f-1(intδ(V)) ≤ intδ(f-1(V)) for each IF setVinY.
Proof . (1) ⇒ (2). Let x (α,β) ∈ cl δ (U), and let B be an IF regular open q -neighborhood of f ( x (α,β) ) in Y . By (1), there exists an IF regular open q -neighborhood A of x (α,β) such that f ( A ) ≤ B . Since x (α,β) ∈ cl δ ( U ) and A is an IF regular open q -neighborhood of x (α,β) , AqU . So f(A)qf(U) . Since f(A) B, Bqf(U) . Then f ( x (α,β) ) ∈ cl δ ( f(U) ). Hence f(clδ(U))) ≤ cl δ ( f ( U )).
(2) ⇒ (3). Let V be an IF set in Y . Then f-1 ( V ) is an IF setin X . By (2), f (cl δ ( f-1 ( V ))) ≤ cl δ ( f ( f-1 ( V ))) ≤ cl δ ( V ).Thus cl δ ( f-1 ( V )) ≤ f-1 (cl δ ( V )).
(3) ⇒ (1). Let x (α,β) be an IF point in X , and let V be an IFregular open q -neighborhood of f ( x (α,β) in Y . Since Vc is anIF regular closed set, Vc is an IF semi-open set. By Theorem1.3, cl( Vc ) = cl δ ( Vc ). Since f ( x(α,β))qV , f ( x (α,β) ) ∉ Vc = cl( Vc ) = cl δ ( Vc ). Therefore x (α,β) f-1 (cl δ ( Vc )). By (3), x (α,β) ∉ cl δ ( f-1 ( Vc )). Then there exists an IF regular open q -neighborhood U of x (α,β) such that
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So U f-1 ( V ), i.e. f ( U ) ≤ V . Hence f is an IF δ -continuous function.
(3) ⇒ (4). Let V be an IF set in Y . By (3), cl δ ( f-1 ( Vc )) ≤ f-1 (cl δ ( Vc )). Thus
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(4) ⇒ (3). Let V be an IF set in Y . Then Vc is an IF set in Y . By the hypothesis, f-1 (int δ ( Vc )) ≤ int δ ( f-1 ( Vc )). Thus
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Hence cl δ ( f-1 ( V )) ̤ f-1 (cl δ ( V )).
The intuitionistic fuzzy δ -continuous function is also characterized in terms of IF δ -open and IF δ -closed sets.
Theorem 2.5. Let f : (X, T) → (Y, T’) be a function. Then the following statements are equivalent:
  • (1)fis an IFδ-continuous function.
  • (2)f-1(A) is an IFδ-closed set for each IFδ-closed setAinX.
  • (3)f-1(A) is an IFδ-open set for each IFδ-open setAinX.
Proof . (1) ⇒ (2). Let A be an IF δ -closed set in X . Then A = cl δ ( A ). By Theorem 2.4, cl δ ( f-1 ( A )) ≤ f-1 (cl δ ( A )) = f-1 ( A ). Hence f-1 ( A ) = cl δ ( f-1 ( A )). Therefore, f-1 ( A ) is an IF δ -closed set.
(2) ⇒ (3). Trivial.
(3) ⇒ (1). Let x (α,β) be an IF point in X , and let V be an IF regular open q -neighborhood of f ( x (α,β) ). By Corollary 1.2, V is an IF δ -open set. By the hypothesis, f-1 ( V ) is an IF δ -open set. Since x(α,β)qf-1 ( V ), by Lemma 2.2, we have that f-1 ( V ) is an IF δ -neighborhood of x (α,β) . Therefore, there exists an IF regular open q -neighborhood U of x (α,β) such that U f-1 ( V ). Hence f ( U ) ≤ V .
The intuitionistic fuzzy δ -continuous function is also characterized in terms of IF δ -neighborhoods.
Theorem 2.6. A function f : (X, T) → (Y, T’) is IF δ -continuous if and only if for each IF point x (α,β) of X and each IF δ -neighborhood N of f ( x (α,β) ), the IF set f-1 ( N ) is an IF δ -neighborhood of x (α,β) .
Proof . Let x (α,β) be an IF point in X , and let N be an IF δ -neighborhood of f ( x (α,β) ). Then there exists an IF regular open q -neighborhood V of f ( x (α,β) ) such that V N . Since f is an an IF δ -continuous function, there exists an IF regular open q-neighborhood U of x (α,β) such that f U ) ≤ V . Thus, U f-1 ( V ) ≤ N . Hence f-1 ( N ) is an IF δ -neighborhood of x (α,β) .
Conversely, let x (α,β) be an IF point in X , and V an IF regular open q -neighborhood of f ( x (α,β) ). Then V is an IF δ -neighborhood of f ( x (α,β) ). By the hypothesis, f-1 ( V ) is an IF δ -neighborhood of x (α,β) . By the definition of IF δ -neighborhood, there exists an IF regular open q -neighborhood U of x (α,β) such that U f-1 ( V ). Thus f ( U ) ≤ V . Hence f is an IF δ -continuous function.
Theorem 2.7. Let f : (X, T) → (Y, T’) be a bijection. Then the following statements are equivalent:
  • (1) f is an IFδ-continuous function.
  • (2) intδ(f(U)) ≤f(intδ(U)) for each IF setUinX.
Proof . (1) ⇒ (2). Let U be an IF set in X . Then f ( U ) is an IF set in Y . By Theorem 2.4, f-1 int δ ( f ( U ))) ≤ int δ ( f-1 ( f ( U ))). Since f is one-to-one,
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Since f is onto,
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(2) ⇒ (1). Let V be an IF set in Y . Then f-1 ( V ) is an IF set in X . By the hypothesis, int δ ( f ( f-1 ( V ))) ≤ f (int δ ( f-1 ( V ))). Since f is onto,
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Since f is one-to-one,
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Hence by Theorem 2.4, f is an IF δ -continuous function.
Recall that a function f : (X, T) → (Y, T’) is said to be fuzzy weakly δ-continuous if for each fuzzy point xα , in X and each fuzzy open q -neighborhood V of f ( xα ), there exists an fuzzy open q -neighborhood U of xα , such that f (int(cl( U ))) ≤ cl( V ) (See [14] ). We define a similar definition in the intuitionistic fuzzy topological spaces as follows.
Definition 2.8. A function f : (X, T) → (Y, T’) is said to be intuitionistic fuzzy weakly δ-continuous if for each intuitionistic fuzzy point x (α,β) in X and each intuitionistic fuzzy open q -neighborhood V of f ( x (α,β) ), there exists an intuitionistic fuzzy open q -neighborhood U of x (α,β) such that
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Theorem 2.9. Let f : (X, T) → (Y, T’) be a function. Then the following statements are equivalent:
  • (1)fis an IF weaklyδ-continuous function.
  • (2)f(clδ(A)) ≤ clθ(f(A)) for each IF setAinX.
  • (3) clδ(f-1(B)) ≤f-1(clθ(B)) for each IF setBinY.
  • (4)f-1(intθ(B)) ≤ intδ(f-1(B)) for each IF setBinY.
Proof . (1) ⇒ (2): Let x (α,β) ∈ cl δ ( A ), and let V be an IF open q -neighborhood of f ( x (α,β) ) in Y . Since f is an IF weakly δ -continuous function, there exists an IF open q -neighborhood U of x (α,β) such that f (int(cl( U ))) ≤ cl( V ). Since int(cl( V )) is an IF regular open q -neighborhood of x (α,β) and x (α,β) ∈ cl δ ( A ), we have Aq int(cl( V )). Thus f ( A ) qf (int(cl( V ))). Since f (int(cl( V ))) ≤ cl( V ), we have f ( A ) q cl( V ). Thus f ( x (α,β) ) ∈ cl θ ( f ( A )). Hence f (cl δ (A)) ≤ cl θ ( f ( A )).
(2) ⇒ (3): Let B be an IF set in Y . Then f-1 ( B ) is an IF set in X . By (2), f cl δ ( f-1 ( B ))) ≤ cl θ ( f ( f-1 ( B ))) ≤ cl θ ( B ). Hence cl δ ( f-1 ( B )) ≤ f-1 (cl θ ( B )).
(3) ⇒ (1): Let x (α,β) be an IF point in X , and let V be an IF open q -neighborhood of f ( x (α,β) ) in Y . Since cl( V ) ≤ cl( V ),
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Thus f ( x (α,β) ) ∉ cl θ ((cl( V )) c ). By (3), f ( x (α,β) ) ∉ cl δ ( f-1 ((cl( V )) c )). Then there exists an intuitionistic fuzzy regular open q -neighborhood U of x (α,β) such that
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Thus int(cl( U )) ≤ f-1 (cl( V )). Therefore, there exists an IF open q -neighborhood U of x (α,β) such that f (int(cl( U ))) ≤ cl( V ). Hence f is an IF weakly δ -continuous function.
(3) ⇒ (4): Let B be an IF set in Y . Then Bc is an IF set in Y . By (3), cl δ ( f-1 ( Bc )) ≤ f-1 (cl θ ( Bc )). Hence we have int δ ( f-1 ( B )) = (cl θ ( f-1 ( Bc ))) ≥ ( f-1 (cl θ ( Bc ))) c = int θ ( f-1 ( B )).
(4) ⇒ (3): Similarly.
Theorem 2.10. A function f : (X, T) → (Y, T’) is IF weakly δ -continuous if and only if for each IF point x (α,β) in X and each IF θ -neighborhood N of f ( x (α,β) ), the IF set f-1 N ) is an IF δ -neighborhood of x (α,β) .
Proof . Let x (α,β) be an IF point in X , and let N be an IF θ -neighborhood of f ( x (α,β) ) in Y . Then there exists an IF open q -neighborhood V of f ( x (α,β) ) such that cl( V ) ≤ N . Since f is an IF weakly δ -continuous function, there exists an IF open q -neighborhood U of x (α,β) such that f (int(cl( U )) x (α,β) cl( V ). Since cl( V ) x (α,β) N , int(cl( U )) x (α,β) f-1 ( N ). Hence f-1 ( N ) is an IF δ -neighborhood of x (α,β) .
Conversely, let x (α,β) be an IF point in X and let V be an IF open q -neighborhood of f ( x (α,β) ). Since cl( V ) ≤ cl( V ), cl( V ) is an IF θ -neighborhood of f ( x (α,β) ). By the hypothesis, f-1 (cl( V )) is an IF θ -neighborhood of x (α,β) . Then there exists an IF open q -neighborhood U of x (α,β) such that int(cl( V )) ≤ f-1 (cl( V )). Thus int(cl( V )) ≤ f-1 (cl( V )). Hence f is IF almost strongly δ -continuous.
Theorem 2.11. Let f : (X, T) → (Y, T’) be an IF weakly δ -continuous function. Then the following statements are true:
  • (1)f-1(V) is an IFθ-closed set inXfor each IFδ-closed setVinY.
  • (2)f-1(V) is an IFθ-open set inXfor each IFδ-open setVinY.
Proof . (1) Let B be an IF θ -closed set in Y . Then cl θ ( B ) = B . Since f is an IF weakly δ -continuous function, by Theorem 2.9, cl δ ( f-1 ( B )) ≤ f-1 (cl θ ( B )) = f-1 ( B ). Hence f-1 ( B ) is an IF δ -closed set in X .
(2) Trivial.
Theorem 2.12. Let f : (X, T) → (Y, T’) be a bijection. Then the following statements are equivalent:
  • (1)fis an IF weaklyδ-continuous function.
  • (2) intθ(f(A)) ≤fintδ(A)) for each IF setAinX.
Proof . (1) ⇒ (2): Let A be an IF set in X . Then f ( A )is an IF set in Y . By Theorem 2.9-(4), f-1 (int θ ( f ( A ))) ≤ int δ ( f-1 ( f ( A ))). Since f is one-to-one,
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Since f is onto,
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Hence int θ ( f ( A )) ≤ f(int δ ( A )).
(2) ⇒ (1): Let B be an IF set in Y . Then f -1 ( B ) is an IF set in X . By (2) int θ ( f ( f -1 ( B ))) ≤ f (int δ ( f -1 ( B ))). Since f is onto,
  • intθ(B) = intθ(f(f-1(B)) ≤f(intδ(f-1(B)):
f is one-to-one,
  • f-1(intθ(B) ≤f-1(f(intδ(f-1(B))) = intδ(<(f-1(B)).
By Theorem 2.9, f is an IF weakly δ-continuous function.
3. IF Almost Continuous and Almost Strongly θcontinuous Functions
Definition 3.1 ( [7] ). A function f : ( X, T ) → ( Y, T’ ) is said to be intuitionistic fuzzy almost continuous if for any intuitionistic fuzzy regular open set V in Y , f-1 ( V ) is an intuitionistic fuzzy open set in X .
Theorem 3.2 ( [12] ). A function f : ( X, T ) → ( Y, T’ ) is IF almost continuous if and only if for each IF point x (α,β) in X and for any IF open q -neighborhood V of ( x (α,β) ), there exists an IF open q -neighborhood U of x (α,β) such that
  • f(U)≤ int(cl(V)).
Theorem 3.3. Let f : ( X, T ) → ( Y, T’ ) be a function. Then the following statements are equivalent:
(1) f is an IF almost continuous function.
(2) f (cl( U )) ≤ cl δ ( f(U) ) for each IF set U in X .
(3) f -1 ( V ) is an IF closed set in X for each IF δ-closed set V in Y .
(4) f -1 ( V ) is an IF open set in X for each IF δ-open set V in Y .
Proof.
(1) ⇒ (2). Let x (α,β) ∈ cl( U ). Suppose that f ( x (α,β) ) ∉ cl δ ( f(U) ). Then there exists an IF open q -neighborhood V of f ( x(α,β)) such that
Lager Image
Since f is an IF almost continuous function, f -1 ( V ) is an IF open set in X . Since Vqf ( x (α,β)) , we have f -1 ( V ) qx (α,β) . Thus f -1 ( V ) is an IF open q -neighborhood of x (α,β) . Since x (α,β) . ∈ cl( U ), by Theorem 1.5, we have f -1 ( V ) qU . Thus f(f-1(V))qf(U) . Since f ( f -1 ( V )) ≤ V, we have Vqf(U) . This is a contradiction. Hence f(cl(U)) ≤ cl δ ( f(U )).
(2) ⇒ (3). Let V be an IF δ-closed set in Y . Then f -1 ( V ) is an IF set in X . By the hypothesis,
  • f(cl(f-1(V)))) ≤ clδ(f(f-1(V))) ≤ clδ(V) =V.
Thus cl( f -1 ( V )) ≤ f -1 ( V ). Hence f -1 ( V ) is an IF closed set in X .
(3) ⇒(4). Let V be an IF δ-open set in Y . Then Vc is an IF δ-closed set in Y . By the hypothesis, f -1 ( Vc ) = ( f -1 ( V )) c is an IF closed set in X . Hence f -1 ( V ) is an IF open set in X .
(4) ⇒ (1). Let x (α,β) be an IF point in X , and let V be an IF open q -neighborhood of f(x(α,β)) in Y . Then int(cl( V )) is an IF regular open q -neighborhood f(x(α,β)) . By Theorem 1.2, int(cl( V )) is an IF δ-open set in Y . By the hypothesis, f -1 (int(cl( V ))) is IF open in X . Since int(cl( V )) qf ( x (α,β) ), we have x(α,β)q ) f -1 (int(cl( V ))). Thus x (α,β) does not belong to the set ( f -1 (int(cl( V )))) c . Put B = ( f -1 (int(cl( V )))) c . Since B is an IF closed set and x (α,β) B = cl( B ), there exists an IF open q -neighborhood U of x (α,β) such that
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Then x (α,β) q U Bc = f -1 (int(cl( V ))). Thus f(U) ≤ int( cl(V )). Hence, f is an IF almost continuous function.
Theorem 3.4. Let f : ( X, T ) → ( Y, T’ ) be a function. Then the following statements are equivalent:
(1) f is an IF almost continuous function.
(2) cl( f -1 ( V )) ≤ f -1 (cl δ ( V )) for each IF set V in Y .
(3) int δ ( f -1 ( V )) ≤ f -1 (int( V )) for each IF set V in Y .
Proof. (1) ⇒ (2). Let V be an IF set in Y . Then f -1 ( V ) is an IF set in X . By Theorem 3.3,
  • f(cl(f-1(V))) ≤ clδ(f(f-1(V))) ≤ clδ(V).
Thus cl( f -1 ( V )) ≤ f -1 (cl δ ( V )).
(2) ⇒ (1). Let U be an IF set in X . Then f(U) is an IF set in Y . By the hypothesis, cl( f -1 ( f(U ))) ≤ f -1 (cl δ ( f(U ))). Then
  • cl(U) ≤ cl(f-1(f(U))) ≤f-1(clδ(f(U))).
Thus f(cl(U )) ≤ cl δ ( f(U )). By Theorem 3.3, f is an IF almost continuous function.
(2) ⇒ (3). Let V be an IF set in Y . Then Vc is an IF set in Y . By the hypothesis, cl( f -1 ( Vc )) ≤ f -1 (cl δ ( Vc )). Thus
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(3) ⇒ (2). Let V be an IF set in Y . Then Vc is an IF set in Y . By the hypothesis, f -1 (int δ ( Vc )) ≤ int( f -1 >( Vc )). Thus
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Hence cl( f -1 ( V )) ≤ f -1 (cl δ ( V )).
Corollary 3.5. A function f : ( X, T ) → ( Y, T’ ) is IF almost continuous if and only if for each IF point x (α,β) in X and each IF δ-neighborhood N of f(x(α,β)) , the IF set f -1 ( N ) is an IF q -neighborhood of x (α,β) .
Proof . Let x (α,β) be an IF point in X , and let N be an IF δ-neighborhood of f(x(α,β)) . Then there exists an IF regular open q -neighborhood V of f(x(α,β)) such that V N . Since f is an IF almost continuous function, there exists an IF open q -neighborhood U of f(x(α,β)) such that f(U) ≤ int(cl( V )) = V N . Thus there exists an IF open set U such that x (α,β) q U f -1 ( N ). Hence f -1 ( N ) is an IF q -neighborhood of x (α,β) .
Conversely, let x (α,β) be an IF point in X , and let V be an IF q -neighborhood of f(x(;)). Then int(cl(V )) is an IF regular open q-neighborhood of f(x(α,β)) . Also, int(cl( V )) is an IF δ-neighborhood of f(x(α,β)) . By the hypothesis, f -1 (int(cl( V )))is an IF q -neighborhood of x (α,β) . Since f -1 (int(cl( V ))) is an IF q -neighborhood of x (α,β) , there exists an IF open q -neighborhood U of x (α,β) such that U f -1 (int(cl( V ))). Thus there exists an IF open q -neighborhood U of x (α,β) such that f(U) ≤ int(cl( V )). Hence f is an IF almost continuous function.
Theorem 3.6. Let f : ( X, T ) → ( Y, T’ ) be a bijection. Then the following statements are equivalent:
(1) f is an IF almost continuous function.
(2) f (int δ ( U )) ≤ int(f(U)) for each IF set U in X .
Proof . Trivial by Theorem 3.4.
Recall that a function f : ( X, T ) → ( Y, T’ ) is said to be a fuzzy almost strongly θ-continuous function if for each fuzzy point xα in X and each fuzzy open q -neighborhood V of f(xα) , there exists an fuzzy open q -neighborhood U of xα such that f (cl( U )) ≤ int(cl( V )) (See [14]).
Definition 3.7 . A function f : ( X, T ) → ( Y, T’ ) is said to be intuitionistic fuzzy almost strongly θ-continuous if for each intuitionistic fuzzy point x (α,β) in X and each intuitionistic fuzzy open q -neighborhood V of f(x(α,β)) , there exists an intuitionistic fuzzy open q -neighborhood U of x(α,β) such that
  • f(cl(U)) ≤ int(cl(V)).
Theorem 3.8. Let f : ( X, T ) → ( Y, T’ ) be a function. Then the following statements are equivalent:
(1) f is an IF almost strongly θ -continuous function.
(2) f (cl θ ( A )) ≤ cl δ ( f(A )) for each IF set A in X .
(3)cl θ ( f -1 ( B )) ≤ f -1 (cl δ ;( B )) for each IF set B in Y .
(4) f -1 (int δ ( B )) ≤ int θ ( f -1 ( B )) for each IF set B in Y .
Proof . (1) ⇒ (2): Let x (α,β) ∈ cl θ ( A ). Suppose f(x(α,β)) ∉ cl δ ( f(A )). Then there exists an IF open q -neighborhood V of f(x(α,β))
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Since f is an IF almost strongly θ continuous function, there exists an IF open q -neighborhood U of x (α,β) such that f (cl( U )) ≤ int(cl( V )) = V . Since f(A) Vc ≤ ( f (cl( U ))) c , we have A ≤ ( f -1 ( f (cl( U )))) c . Thus
Lager Image
Since x (α,β) ∈ cl θ ( A ), we have Aq cl( U ). This is a contradiction.
(2) ⇒ (3): Let B be an IF set in Y . Then f -1 ( B ) is an IF set in X. By (2), f(cl θ ( f -1 ( B ))) ≤ cl θ ( f ( f -1 ( B ))) ≤ cl θ ( B ). Thus we have f (cl θ ( f -1 ( B ))) ≤ cl θ ( f ( f -1 ( B ))) ≤ cl θ ( B ). Hence cl θ ( f -1 ( B )) ≤ f -1 (cl δ ( B )).
(3) ⇒ (4): Let B be an IF set in Y . Then Bc is an IF set in Y . By (3), cl θ ( f -1 ( Bc )) ≤ f -1 (cl δ ( Bc )) for each IF set B in Y . Therefore f -1 (int(B)) = (cl δ ( f -1 ( Bc ))) c ≥ ( f -1 (cl δ ( Bc ))) c = int θ ( f -1 ( B )).
(4) ⇒ (1): Let B be an IF set in Y . Then Bc is an IF set in Y . By (4), f -1 (int δ ( Bc )) ≤ int θ ( f -1 ( Bc )). Thus cl θ ( f -1 ( Bc )) ≤ f -1 (cl δ ( Bc )). Hence f is an IF almost strongly θ -continuous function.
Theorem 3.9. Let f : ( X , T ) → ( Y , T’ ) be a function. Then the following statements are equivalent:
(1) f is an IF almost strongly θ -continuous function.
(2) The inverse image of every IF δ-closed set in Y is an IF θ -closed set in X .
(3) The inverse image of every IF δ-open set in Y is an IF θ -open set in X .
(4) The inverse image of every IF regular open set in Y is an IF θ -open set in X .
Proof . (1) ⇒ (2): Let B be an IF δ-closed set in Y . Then cl δ ( B ) = B . Since f is an IF almost strongly θ -continuous function, by Theorem 3.8, cl θ ( f -1 ( B )) ≤ f -1 cl δ ( B )) = f -1 ( B ). Thus cl θ ( f -1 ( B )) = f -1 ( B ). Hence f -1 ( B ) is an IF δ-closed set in X .
(2) ⇒ (3): Let B be an IF δ-open set in Y . Then Bc is anIF δ-closed set in Y . By (4), f -1 ( Bc ) = ( f -1 ( B )) c is an IF θ -closed set in X . Hence f -1 ( B ) is an IF θ -open set in X .
(3) ⇒ (4): Immediate since IF regular open sets are IF θ -open sets.
(4) ⇒ (1): Let x(α,β) be an IF point in X , and let V be an IF open q -neighborhood of f (x(α,β)). Then int(cl( V )) is an IF regular open q -neighborhood of f (x(α,β)). By (4), f -1 (int(cl( V ))) is an IF θ -open set in X . Then
Lager Image
Put int(cl( V )) = D . Suppose x (α,β) ∈ ( f -1 (int(cl( V )))) c = f -1 ( Dc ). Then
Lager Image
Let f ( x (α,β) ) = y (α0,β0) . Then α 0 ≤ ϓ D ( y ) and β 0 ≥ μ D ( y ). Since V is an IF open set, V ≤ int(cl( V )) = D . Thus μ V ≤ μ D and ϓ u ϓ D . Thus α 0 ϓ V ( y ) and α 0 ≥ μ V ( y ). Since V is an IF open q -neighborhood of f (x (α,β) ), we have f (x (α,β) ) q V . Thus y (α0,β0) V c = ( ϓ V , μ V ). Hence α 0 > ϓ V ( y ) and β 0 < μ V ( y ). This is a contradiction. Therefore there exists an IF open q-neighborhood U of x (α, β) such that
Lager Image
i.e. cl( U ) ≤ f -1 (int(cl( V ))). Then f (cl( U )) ≤ int(cl( V )). Hence f is an IF almost strongly θ -continuous function.
Theorem 3.10. A function f : ( X , T ) → ( Y , T ’) is IF almost strongly θ -continuous if and only if for each IF point x (α,β) in X and each IF δ -neighborhood N of f ( x (α,β) ), the IF set f -1 ( N ) is an IF θ -neighborhood of x (α,β) .
Proof . Let x (α,β) be an IF point in X , and let N be an IF δ -neighborhood of f ( x α,β) ). Then there exists an an IF regular open q -neighborhood V of f ( x (α,β) ) such that V N . Thus int(cl( V )) ≤ N . Since f is an IF almost strongly θ continuous function, there exists an IF open q -neighborhood U of x (α,β) such that f (cl( U )) ≤ int(cl( V )). Thus f (cl( U )) ≤ N . Therefore, there exists an IF open q -neighborhood U of x (α,β) such that cl( U ) ≤ f -1 ( N ). Hence f -1 ( N ) is an IF θ -neighborhood of x (α,β) .
Conversely, let x (α,β) be an IF point in X , and let V be an IF open q -neighborhood of f ( x (α,β) ). Since int(cl( V )) is an IF regular open q -neighborhood of f ( x (α,β) ) and int(cl( V )) ≤ int(cl( V )), int(cl( V )) is an IF δ -neighborhood of f ( x (α,β) ). By the hypothesis, f -1 (int(cl( V ))) is an IF θ -neighborhood of x (α,β) . Then there exists an IF open q -neighborhood U of x (α,β) such that cl( U ) ≤ f -1 (int(cl(V))). Therefore f(cl(U)) ≤ int(cl( V )). Hence f is IF almost strongly θ -continuous.
Theorem 3.11. Let f : ( X, T ) → ( Y, T’ ) be a bijection. Then the following statements are equivalent:
(1) f is an IF almost strongly θ -continuous function.
(2) int δ ( f(A )) ≤ f(int θ ( A )) for each IF set A in X .
Proof. (1) ⇒ (2): Let A be an IF set in X . Then f(A) is an IF set in Y . By Theorem 3.9, f -1 (int θ ( f(A ))) ≤ int θ ( f -1 ( f(A ))). Since f is one-to-one,
  • f-1(intδ(f(A))) ≤ intθ(f-1(f(A))) = intθ(A):
Since f is onto,
  • intδ(f(A)) =f(f-1(intδ(f(A)))) ≤f(intθ(A)):
(2) ⇒ (1): Let B be an IF set in Y . Then f -1 ( B ) is an IF set in X . By (2), int δ ( f -1 ( B ))) ≤ f (int θ ( f -1 ( B ))). Since f is onto,
  • intδ(B) = intδ(f(f-1(B))) ≤f(intθ(f-1(B))):
Since f is one-to-one,
  • f-1(intδ(B)) ≤f-1(f(intθ(f-1(B)))) = intθ(f-1(B)):
By Theorem 3.9, f is an IF almost strongly θ -continuous function.
4. Conclusion
We characterized the intuitionistic fuzzy δ-continuous functions in terms of IF δ-closure and IF δ-interior, or IF δ-open and IF δ-closed sets, or IF δ-neighborhoods.
Moreover, we characterized the IF weakly δ-continuous, IF almost continuous, and IF almost strongly δ-continuous functions in terms of closure and interior.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
References
Atanassov K. T. 1986 “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems 20 (1) 87 - 96    DOI : 10.1016/S0165-0114(86)80034-3
Çoker D. 1996 “An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces,” Journal of Fuzzy Mathematics 4 (4) 749 - 764
Çoker D. 1997 “An introduction to intuitionistic fuzzy topological spaces,” Fuzzy Sets and Systems 88 (1) 81 - 89    DOI : 10.1016/S0165-0114(96)00076-0
Çoker D. 1995 “On intuitionistic fuzzy points,” Notes on Intuitionistic Fuzzy Sets 1 (2) 79 - 84
Sivaraj D. 1986 “Semiopen set characterizations of almostregular spaces,” Glasnik Matematicki Series III 21 (2) 437 - 440
Rama A. A. 1992 “Smooth topological spaces,” Fuzzy Sets and Systems 48 (3) 371 - 375    DOI : 10.1016/0165-0114(92)90352-5
G¨urçay H. , Çoker D. , Es A. H. 1997 “On fuzzy continuity in intuitionistic fuzzy topological spaces,” Journal of Fuzzy Mathematics 5 (2) 365 - 378
Lee S. J. , Lee E. P. 2000 “The category of intuitionistic fuzzy topological spaces,” Bulletin of the Korean Mathematical Society 37 (1) 63 - 76
Lee S. J. , Kim J. T. 2011 “Properties of fuzzy (, s)-semiirresolute mappings in intuitionistic fuzzy topological spaces,” International Journal of Fuzzy Logic and Intelligent Systems 11 (3) 190 - 196    DOI : 10.5391/IJFIS.2011.11.3.190
Hanafy I. M. 2002 “On fuzzy -open sets and fuzzy -continuity in intuitionistic fuzzy topological spaces,” Journal of Fuzzy Mathematics 10 (1) 9 - 19
Mondal T. K. , Samanta S. K. 2002 “On intuitionistic gradation of openness,” Fuzzy Sets and Systems 131 (3) 323 - 336    DOI : 10.1016/S0165-0114(01)00235-4
Hanafy I. M. 2003 “Intuitionistic fuzzy functions,” International Journal of Fuzzy Logic and Intelligent Systems 3 (2) 200 - 205    DOI : 10.5391/IJFIS.2003.3.2.200
Hanafy I. M. , Abd El-Aziz , Salman T. M. 2006 “Intuitionistic fuzzy -closure operator,” Boletn de la Asociacin Matemtica Venezolana 13 (1) 27 - 39
Mukherjee M. N. , Sinha S. P. 1990 “On some near-fuzzy continuous functions between fuzzy topological spaces,” Fuzzy Sets and Systems 34 (2) 245 - 254    DOI : 10.1016/0165-0114(90)90163-Z
Mukherjee M. N. , Sinha S. P. 1991 “Fuzzy θ-closure operator on fuzzy topological spaces,” International Journal of Mathematics and Mathematical Sciences 14 (2) 309 - 314    DOI : 10.1155/S0161171291000364
Lee S. J. , Eoum Y. S. 2010 “Intuitionistic fuzzy -closure and -interior,” Communications of the Korean Mathematical Society 25 (2) 273 - 282    DOI : 10.4134/CKMS.2010.25.2.273
Eom Y. S. , Lee S. J. 2012 “Delta closure and delta interior in intuitionistic fuzzy topological spaces,” International Journal of Fuzzy Logic and Intelligent Systems 12 (4) 290 - 295    DOI : 10.5391/IJFIS.2012.12.4.290
Ganguly S. , Saha S. 1988 “A note on δ-continuity and δ-connected sets in fuzzy set theory,” Simon Stevin (A Quarterly Journal of Pure and Applied Mathematics) 62 (2) 127 - 141