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Intuitionistic Fuzzy Theta-Compact Spaces
Intuitionistic Fuzzy Theta-Compact Spaces
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Sep, 13(3): 224-230
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 : July 19, 2013
  • Accepted : September 11, 2013
  • Published : September 25, 2013
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Yeon Seok Eom
Seok Jong Lee

Abstract
In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy θ -compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzy θ -compactness is strictly weaker than intuitionistic fuzzy compactness. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzy θ -compactness in the original intuitionistic fuzzy topology. This characterization shows that intuitionistic fuzzy θ -compactness can be related to an appropriated notion of intuitionistic fuzzy convergence.
Keywords
1. Introduction
The concept of an intuitionistic fuzzy set as a generalization of fuzzy sets was introduced by Atanassov [1] . Coker and his colleagues [2 ? 4] introduced an intuitionistic fuzzy topology using intuitionistic fuzzy sets.
Many researchers studied continuity and compactness in fuzzy topological spaces and intuitionistic fuzzy topological spaces [5 ? 8] . Recently, Hanafy et al. [9] introduced an intuitionistic fuzzy θ -closure operator and intuitionistic fuzzy θ -continuity.
In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy θ -compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzy θ -compactness is strictly weaker than intuitionistic fuzzy compactness. Moreover, we show that the sufficient condition in Theorem 4.5 holds for intuitionistic fuzzy θ -compact spaces; however, in general, it fails for intuitionistic fuzzy compact spaces. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to the intuitionistic fuzzy θ -compactness in the original intuitionistic fuzzy topology described in Theorem 4.6. This characterization shows that the intuitionistic fuzzy θ -compactness can be related to an appropriated notion of intuitionistic fuzzy convergence.
2. Preliminaries
Let X and I denote a nonempty set and unit interval [0, 1], respectively. An intuitionistic fuzzy set A in X is an object of the form
  • A= (μA,γA),
where the functions μA : X I and γA : X I denote the degree of membership and the degree of non-membership, respectively, and μA + γA ≤ 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 intuitionistic fuzzy is abbreviated as IF.
Definition 2.1. [1] Let X denote a nonempty set and let intuitionistic fuzzy sets A and B be of the form A = ( μA , γA ), B = ( μB , γB ). Then,
  • (1)A≤BiffμA(x) ≤μB(x) andγA(x) ≥γB(x) for allx∈X,
  • (2)A=BiffA≤BandB≤A,
  • (3)Ac= (γA,μA),
  • (4)A∩B= (μA∧μB,γA∨γB),
  • (5)A∪B= (μA∨μB,γA∧γB),
Lager Image
Definition 2.2. [2] An intuitionistic fuzzy topology on X is a family Τ of intuitionistic fuzzy sets in X that satisfy the following axioms.
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  • (2)G1∩G2∈Τfor anyG1,G2∈Τ,
  • (3) ⋃Gi∈Τfor any {Gi:i∈J} ⊆Τ.
In this case, the pair ( X, Τ ) is called an intuitionistic fuzzy topological space and any intuitionistic fuzzy set in Τ is known as an intuitionistic fuzzy open set in X .
Definition 2.3. [2] Let ( X, Τ ) and A denote an intuitionistic fuzzy topological space and intuitionistic fuzzy set in X , respectively. Then, the intuitionistic fuzzy interior of A and the intuitionistic fuzzy closure of A are defined by
  • cl(A) = ⋂{K|A≤K,Kc∈Τ}
and
  • int(A) = ⋃{G|G≤A,G∈Τ}
Theorem 2.4. [2] For any IF set A in an IF topological space ( X, Τ ), we have
  • cl(Ac) = (int(A))cand int(Ac) = (cl(A))c.
Definition 2.5. [3 , 4] Let α,β ∈ [0, 1] and α + β ≤ 1. An intuitionistic fuzzy point x (α,β) of X is an intuitionistic fuzzy set in X defined by
Lager Image
In this case, x,α, and β are called the support, value, and nonvalue of x (α,β) , respectively. An intuitionistic fuzzy point x (α,β) is said to belong to an intuitionistic fuzzy set A = ( μA , γA ) in X , denoted by x (α,β) A , if α μA ( x ) and β γA ( x ).
Remark 2.6. If we consider an IF point x (α,β) as an IF set, then we have the relation x (α,β) A if and only if x (α,β) A .
Definition 2.7. [4 , 10] Let ( X,Τ ) denote an intuitionistic fuzzy topological space.
  • (1) An intuitionistic fuzzy pointx(α,β)is said to bequasicoincidentwith the intuitionistic fuzzy setU= (μU,γU), denoted byx(α,β)qU, ifα>γU(x) or β <μU(x).
  • (2) LetU= (μU,γU) andV= (μV,γV) denote two intuitionistic fuzzy sets inX. Then,UandVare said to bequasi-coincident, denoted byUqV, if there exists an elementx∈Xsuch thatμU(x) >γV(x) orγU(x) <μV(x).
The word ‘not quasi-coincident’ will be abbreviated as
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herein.
Proposition 2.8. [4] Let U, V , and x (α,β) denote IF sets and an IF point in X , respectively. Then,
Lager Image
Lager Image
Lager Image
Lager Image
Definition 2.9. [4] Let ( X,Τ ) denote an intuitionistic fuzzy topological space and let x (α,β) denote an intuitionistic fuzzy point in X . An intuitionistic fuzzy set A is said to be an intuitionistic fuzzy ϵ-neighborhood ( q-neighborhood ) of x (α,β) if there exists an intuitionistic fuzzy open set U in X such that x (α,β) U A ( x (α,β) qU A , respectively).
Theorem 2.10. [10] Let x (α,β) and U = ( μU , γU ) denote an IF point in X and an IF set in X , respectively. Then, x (α,β) ∈ cl( U ) if and only if UqN , for any IF q -neighborhood N of x (α,β) .
Definition 2.11. [9] An intuitionistic fuzzy point x (α,β) is said to be an intuitionistic fuzzy θ-cluster point of an intuitionistic fuzzy set A if for each intuitionistic fuzzy q -neighborhood U of x (α,β) , Aq cl( U ). The set of all intuitionistic fuzzy θ -cluster points of A is called intuitionistic fuzzy θ-closure of A and is denoted by cl θ ( A ). An intuitionistic fuzzy set A is called an intuitionistic fuzzy θ-closed set if A = cl θ ( A ). The complement of an intuitionistic fuzzy θ -closed set is said to be an intuitionistic fuzzy θ-open set .
Definition 2.12. [11] Let ( X,Τ ) and U denote an intuitionistic fuzzy topological space and an intuitionistic fuzzy set in X , respectively. The intuitionistic fuzzy θ-interior of U is denoted and defined by
  • intθ(U) = (clθ(Uc))c.
Definition 2.13. [2] Let ( X,Τ ) and ( Y,U ) denote two intuitionistic fuzzy topological spaces and let f : X Y denote a function. Then, f is said to be intuitionistic fuzzy continuous if the inverse image of an intuitionistic fuzzy open set in Y is an intuitionistic fuzzy open set in X .
Definition 2.14. [2] An intuitionistic fuzzy topological space ( X,Τ ) is said to be intuitionistic fuzzy compact if every open cover of X has a finite subcover.
Definition 2.15. [9] A function f : X Y is said to be intuitionistic fuzzy θ-continuous if for each intuitionistic fuzzy point x (a,b) in X and each intuitionistic fuzzy open q -neighborhood V of f ( x (a,b) ), there exists an intuitionistic fuzzy open q -neighborhood U of x (a,b) such that f (cl( U )) ≤ cl( V ).
Proposition 2.16. [12] Let f : ( X,Τ ) → ( Y,T' ) and x (α,β) denote a function and an IF point in X , respectively.
  • (1) Iff(x(α,β))qV, thenx(α,β)qf–1(V) for any IF setVinY.
  • (2) Ifx(α,β)qU, thenf(x(α,β))qf(U) for any IF setUinX.
Remark 2.17. Intuitionistic fuzzy sets have some different properties compared to fuzzy sets, and these properties are shown in the subsequent examples.
  • 1.x(α,β)∈A∪B⇏x(α,β)∈Aorx(α,β)∈B.
  • 2.x(α,β)qAandx(α,β)qB⇏x(α,β)q(A∩B).
Thus, we have considerably different results in generalizing concepts of fuzzy topological spaces to the intuitionistic fuzzy topological space.
Example 2.18. Let A, B denote IF sets on the unit interval [0, 1] defined by
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In addition, let x = ¼, ( α,β ) = (¼,½). Then, x (α,β) A B . However, x (α,β) A and x (α,β) B .
Example 2.19. Let A, B denote IF sets on the unit interval [0, 1] defined by
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In addition, let x = ¼, ( α,β ) = (½,¼). Then, x (α,β) qA and x (α,β) qB ; however,
Lager Image
For the notions that are not mentioned in this section, refer to [11] .
3. Intuitionistic Fuzzy θ-Irresolute and Weakly θ-Continuity
Definition 3.1. Let ( X,Τ ) and ( Y,U ) be IF topological spaces. A mapping f : ( X,Τ ) → ( Y,U ) is said to be intuitionistic fuzzy θ-irresolute if the inverse image of each IF θ -open set in Y is IF θ -open in X .
Theorem 3.2. Let ( X,Τ ) and ( Y,U ) be IF topological spaces. Let Τθ be an IF topology on X generated using the subbase of all the IF θ -open sets in X , and let Uθ be an IF topology on Y generated using the subbase of all the IF θ -open sets in Y . Then a function f : ( X,Τ ) → ( Y,U ) is IF θ -irresolute if and only if f : ( X,Τθ ) → ( Y,Uθ ) is IF continuous.
Proof. Trivial.
Recall that a fuzzy set A is said to be a fuzzy θ-neighborhood of a fuzzy point xα if there exists a fuzzy closed q -neighborhood U of xα , such that
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[13] .
Definition 3.3. An intuitionistic fuzzy set A is said to be an intuitionistic fuzzy θ-neighborhood of intuitionistic fuzzy point x (α,β) if there exists an intuitionistic fuzzy open q -neighborhood U of x (α,β) such that cl( U ) ≤ A .
Recall that a function f : ( X,Τ ) → ( Y,T' ) is said to be a fuzzy weakly θ-continuous function if for each fuzzy point xα in X and each fuzzy open q -neighborhood V of f ( xα ), there exists a fuzzy open q -neighborhood U of xα such that f ( U ) ≤ cl( V ) [13] .
Definition 3.4. A function f : ( X,Τ ) → ( Y,T' ) is said to be intuitionistic fuzzy weakly 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 ( U ) ≤ cl( V ).
Theorem 3.5. A function f : ( X,Τ ) → ( Y,T' ) is IF weakly θ -continuous if and only if for each IF point x(α,β) in X and each IF open θ -neighborhood N of f ( x(α,β) ) in Y , f –1 ( N ) is an IF q -neighborhood of x(α,β) .
Proof. Let f be an IF weakly θ -continuous function, and let x(α,β) be an IF point in X . 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 IF weakly θ -continuous, there exists an IF q -neighborhood U of x(α,β) such that f ( U ) ≤ cl( V ) ≤ N . Thus U f –1 ( N ). Therefore, there exists an IF q -neighborhood U of x(α,β) such that 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 open q -neighborhood of f ( x(α,β) ). Then cl( V ) is an IF θ -neighborhood of f ( x(α,β) ). By the hypothesis, f –1 (cl( V )) is an an IF q -neighborhood of x(α,β) . Then there exists an IF open set U such that x(α,β) qU f –1 (cl( V )). Thus f ( U ) ≤ cl( V ). Therefore there exists an IF open q -neighborhood U of x(α,β) such that f ( U ) ≤ cl( V ). Hence f is an IF weakly θ -continuous function.
Theorem 3.6. If a function f : ( X,Τ ) → ( Y,T' ) is IF weakly θ -continuous, then
  • (1)f(cl(A)) ≤ clθ(f(A)) for each IF setAinX,
  • (2)f(cl(int(cl(f–1(B))))) ≤ clθ(B) for each IF setBinY.
Proof. (1) Let x(α,β) ∈ cl( A ), and let V be an IF open q -neighborhood of f ( x(α,β) ). Since f is IF weakly θ -continuous, there exists an IF open q -neighborhood U of x(α,β) such that f ( U ) ≤ cl( V ). Since x(α,β) ∈ cl( A ), UqA . Thus f ( U ) qf ( A ). Since f ( U ) ≤ cl( V ), we have cl( V ) qf ( A ). Thus for each IF open q -neighborhood V of f ( x(α,β) ), cl( V ) qf ( A ). Hence f ( x(α,β) ) ∈ cl θ ( f ( A )).
(2) Let B be an IF set in Y and x(α,β) ∈ cl(int(cl( f –1 ( B )))). Let V be an IF open q -neighborhood of f ( x(α,β) ). Since f is IF weakly θ -continuous, there exists an IF open q -neighborhood U of x(α,β) such that f ( U ) ≤ cl( V ). Since int(cl( f –1 ( B ))) ≤ cl( f –1 ( B )),
  • cl(int(cl(f–1(B)))) ≤ cl(cl(f–1(B))) = cl(f–1(B)).
Since x(α,β) ∈ cl(int(cl( f –1 ( B )))), x(α,β) ∈ cl( f –1 ( B )). Thus f –1 ( B ) qU , or Bqf ( U ). Since f ( U ) ≤ cl( V ), we have cl( V ) qB . Therefore f ( x(α,β) ) ∈ cl θ ( B ). Hence we obtain f (cl(int(cl( f –1 ( B ))))) ≤ cl θ ( B ), for each IF set B in Y .
Theorem 3.7. Let f : ( X,Τ ) → ( Y,T' ) be a function. Then the following statements are equivalent:
  • (1)fis an IF weaklyθ-continuous function.
  • (2) For each IF open setUwithx(α,β)qf–1(U),x(α,β)qint(f–1(cl(U))).
Proof. (1) ⇒ (2). Let f be an IF weakly θ -continuous function, and let U be an IF open set with x(α,β) qf –1 ( U ). Then f ( x(α,β) ) qU . By the definition of IF weakly θ -continuous, there exists an IF open q -neighborhood V of x(α,β) such that f ( V ) ≤ cl( U ). Thus V f –1 (cl( U )), i.e.
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Therefore, x(α,β) ∉ cl(( f –1 (cl( U ))) c ) = (int( f –1 (cl( U )))) c . Hence we have x(α,β) q (int( f –1 (cl( U )))).
(2) ⇒ (1). Let the condition hold, and let x(α,β) be any IF point in X and V an IF open q -neighborhood of f ( x(α,β) ). Then x(α,β) qf –1 ( V ). By the hypothesis,
  • x(α,β)qint(f–1(cl(V))).
Put U = int( f –1 (cl( V ))). Then U is an IF open q -neighborhood of x(α,β) . Since int( f –1 (cl( V ))) ≤ f –1 (cl( V )),
  • f(int(f–1(cl(V)))) ≤f(f–1(cl(V))) ≤ cl(V).
Thus f ( U ) ≤ cl( V ). Therefore there exists an IF open q -neighborhood U of x(α,β) such that f ( U ) ≤ cl( V ). Hence f is an IF weakly θ -continuous function.
4. Intuitionistic Fuzzy θ-Compactness
Definition 4.1. A collection { Gi | i I } of intuitionistic fuzzy θ -open sets in an intuitionistic fuzzy topological space ( X,Τ ) is said to be an intuitionistic fuzzy θ-open cover of a set A if A ˅ { Gi | i I }.
Definition 4.2. An intuitionistic fuzzy topological space ( X,Τ ) is said to be intuitionistic fuzzy θ-compact if every intuitionistic fuzzy θ -open cover of X has a finite subcover.
Definition 4.3. A subset A of an intuitionistic fuzzy topological space ( X,Τ ) is said to be intuitionistic fuzzy θ-compact if for every collection { Gi | i I } of intuitionistic fuzzy θ -open sets of X such that A ˅ { Gi | i I }, there is a finite subset I 0 of I such that A ˅ { Gi | i I 0 }.
Remark 4.4. Since every IF θ -open set is IF open, it follows that every IF compact space is IF θ -compact.
Theorem 4.5. An IF topological space ( X,Τ ) is IF θ -compact if and only if every family of IF θ -closed subsets of X with the finite intersection property has a nonempty intersection.
Proof. Let X be IF θ -compact and let F = { Fi | i I } denote any family of IF θ -closed subsets of X with the finite intersection property. Suppose that
Lager Image
Then,
Lager Image
is an IF θ -open cover of X . Since X is IF θ -compact, there is a finite subset I 0 of I such that
Lager Image
This implies that
Lager Image
which contradicts the assumption that F has a finite intersection property. Hence,
Lager Image
Let g = { Gi | i I } denote an IF θ -open cover of X and consider the family
Lager Image
of an IF θ -closed set. Since g is a cover of X ,
Lager Image
Hence, g' does not have the finite intersection property, i.e., there are finite numbers of IF θ -open sets { G 1 , G 2 , … , Gn } in g such that
Lager Image
This implies that { G 1 , G 2 , … , Gn } is a finite subcover of X in g . Hence, X is IF θ -compact.
Theorem 4.6. Let ( X,Τ ) denote an IF topological space and Τθ denote the IF topology on X generated using the subbase of all IF θ -open sets in X . Then, ( X,Τ ) is IF θ -compact if and only if ( X,Τθ ) is IF compact.
Proof. Let ( X,Τθ ) be IF compact and let g = { Gi | i I } denote an IF θ -open cover of X in T . Since for each i I, Gi Τθ, g is an IF open cover of X in Τθ . Since ( X,Τθ ) is IF compact, g has a finite subcover of X . Hence, ( X,Τ ) is IF θ -compact.
Let ( X,Τ ) be IF θ -compact and let g = { Gi | Gi Τθ, i I } denote an IF open cover of X in Τθ . Since for each i I, Gi Τθ , Gi is an IF θ -open set in ( X,Τ ). Therefore, g is an IF θ -open cover of X in T . Since ( X,Τ ) is IF θ -compact, g has a finite subcover of X . Hence, ( X,Τθ ) is IF compact.
Theorem 4.7. Let A be an IF θ -closed subset of an IF θ -compact space X . Then, A is also IF θ -compact.
Proof. Let A denote an IF θ -closed subset of X and let g = { Gi | i I } denote an IF θ -open cover of A . Since Ac is an IF θ -open subset of X, g = { Gi | i I } ∪ Ac is an IF θ -open cover of X . Since X is IF θ -compact, there is a finite subset I 0 of I such that
Lager Image
Hence, A is IF θ -compact relative to X .
Theorem 4.8. An IF topological space ( X,Τ ) is IF θ -compact if and only if every family of IF closed subsets of X in Τθ with the finite intersection property has a nonempty intersection.
Proof. Trivial by Theorem 4.5.
Theorem 4.9. Let ( X,Τ ) and ( Y,U ) denote IF topological spaces. Let Τθ denote an IF topology on X generated by the subbase of all IF θ -open sets in X and let Uθ denote an IF topology on Y generated by the subbase of all IF θ -open sets in Y . Then, a function f : ( X,Τ ) → ( Y,U ) is IF θ -irresolute if and only if f : ( X,Τθ ) → ( Y,Uθ ) is IF continuous.
Proof. Trivial.
Recall that a function f : ( X,Τ ) → ( Y,T' ) is said to be intuitionistic fuzzy strongly θ-continuous if for each IF point x(α,β) in X and for each IF open q -neighborhood V of f ( x(α,β) ), there exists an IF open q -neighborhood U of x(α,β) such that f (cl( U )) ≤ V ( [9] ).
  • Theorem 4.10.(1) An IF stronglyθ-continuous image of an IFθ-compact set is IF compact.
  • (2) Let (X,Τ) and (Y,U) denote IF topological spaces and letf: (X,Τ) → (Y,U) be IFθ-irresolute. If a subsetAofXis IFθ-compact, then imagef(A) is IFθ-compact.
Proof. (1) Let f : ( X,Τ ) → ( Y,U ) denote an IF strongly θ -continuous mapping from an IF θ -compact space X onto an IF topological space Y . Let g = { Gi | i I } be an IF open cover of Y . Since f is an IF strongly θ -continuous function, f : ( X,Τθ ) → ( Y,U ) is an IF continuous function (Theorem 4.2 of [11] ). Therefore, { f –1 ( Gi ) | i I } is an IF θ -open cover of X . Since X is IF θ -compact, there is a finite subset I 0 of I such that
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Since f is onto, { Gi | i I 0 } is a finite subcover of Y . Hence, Y is IF compact.
(2) Let g = { Gi | i I } be an IF θ -open cover of f ( A ) in Y . Since f is an IF θ -irresolute, for each Gi , f –1 ( Gi ) is an IF θ -open set. Moreover, { f –1 ( Gi ) | i I } is an IF θ -open cover of A . Since A is IF θ -compact relative to X , there exists a finite subset I 0 of I such that A ˅ { f –1 ( Gi ) | i I 0 }. Therefore, f ( A ) ≤ ˅ { Gi | i I 0 }. Hence, f ( A ) is IF θ -compact relative to Y .
Theorem 4.11. Let A and B be subsets of an IF topological space ( X,Τ ). If A is IF θ -compact and B is IF θ -closed in X , then A B is IF θ -compact.
Proof. Let g = { Gi | i I } be an IF θ -open cover of A ˄ B in X . Since Bc is IF θ -open in X , ( ˅ { Gi | i I }) ∨ Bc is an IF θ -open cover of A . Since A is IF θ -compact, there is a finite subset I 0 of I such that A ≤ ( ˅ { Gi | i I 0 }) ∨ Bc . Therefore, A B ≤ ( ˅ { Gi | i I 0 }). Hence, A B is IF θ -compact.
5. Conclusion
We introduced IF θ -irresolute and weakly θ -continuous functions, and intuitionistic fuzzy θ -compactness in intuitionistic fuzzy topological spaces. We showed that intuitionistic fuzzy θ -compactness is strictly weaker than intuitionistic fuzzy compactness. Moreover, we showed that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzy θ -compactness in the original intuitionistic fuzzy topology.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
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