Intuitionistic Fuzzy ？？continuous Functions
 Author: Eom Yeon Seok, Lee Seok Jong
 Organization: Eom Yeon Seok; Lee Seok Jong
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue4, p336~344, 25 Dec 2013

ABSTRACT
In this paper, we characterize the intuitionistic fuzzy 𝛿continuous, intuitionistic fuzzy weakly 𝛿continuous, intuitionistic fuzzy almost continuous, and intuitionistic fuzzy almost strongly 𝜃continuous functions in terms of intuitionistic fuzzy 𝛿closure and interior or 𝜃closure and interior.

KEYWORD
Intuitionistic fuzzy ？？continuous , Weakly ？？continuous , Almost continuous , Almost strongly ？？continuous

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 𝜃closure as a generalization of the concept of fuzzy 𝜃closure 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 𝜃interior and 𝛿closure in intuitionistic fuzzy topological spaces.
In this paper, we characterize the intuitionistic fuzzy 𝛿continuous, intuitionistic fuzzy weakly 𝛿continuous, intuitionistic fuzzy almost continuous, and intuitionistic fuzzy almost strongly 𝜃continuous functions in terms of intuitionistic fuzzy 𝛿closure and interior, or 𝜃closure and interior.
Let
X be a nonempty set andI the unit interval [0, 1]. Anintuitionistic fuzzy set A inX is an object of the formA = (𝜇_{A} , 𝛾_{A} ), 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_{A} inX is an intuitionistic fuzzy set of the form (𝜇_{A} , 1 − 𝜇_{A} ).Throughout this paper,
I (X ) denotes the family of all intuitionistic fuzzy sets inX , 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 set A is IF semiopen in X. (2) A ≤ cl(int(A)).
Corollary 1.2 ( [17]). IfU is an IF regular open set, thenU is an IF 𝛿open set.Theorem 1.3 ( [17]). For any IF semiopen setA , we have cl(A ) = cl_{𝛿}(A ).Lemma 1.4 ( [17]). (1) For any IF setU in an IF topological space (X , 𝛵), int(cl(U )) is an IF regular open set.(2) For any IF open set
U in an IF topological space (X , 𝛵) such thatx _{(𝛼,𝛽)}qU , int(cl(U )) is an IF regular openq neighborhood ofx _{(𝛼,𝛽)}.Theorem 1.5 ( [12]). Letx _{(𝛼,𝛽)} be an IF point inX , andU = (𝜇_{U} , 𝛾_{U} ) an IF set inX . Thenx _{(𝛼,𝛽)} ∈ cl(U ) if and only ifUqN , for any IFq neighborhoodN ofx _{(𝛼,𝛽)}.2. Intuitionistic Fuzzy ？？continuous and Weakly ？？continuous Functions
Recall that a fuzzy set
N in (X , 𝛵) is said to be afuzzy 𝛿neighborhood of a fuzzy pointx _{𝛼} if there exists a fuzzy regular openq neighborhoodV ofx 𝛼 such that or equivalentlyV ≤N (See [14]). Now, we define a similar definition in the intuitionistic fuzzy topological spaces.Definition 2.1. An intuitionistic fuzzy setN in (X , 𝛵) is said to be anintuitionistic fuzzy 𝛿neighborhood of an intuitionistic fuzzy pointx _{(𝛼,𝛽)} if there exists an intuitionistic fuzzy regular openq neighborhoodV ofx _{(𝛼,𝛽)} such thatV ≤N .Lemma 2.2. An IF setA is an IF 𝛿open set in (X , 𝛵) if and only if for any IF pointx _{(𝛼,𝛽)} withx _{(𝛼,𝛽) }qA ,A is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}.Proof . LetA be an IF 𝛿open set in (X , 𝛵) such that x_{(𝛼,𝛽)}qA . Thenx _{(𝛼,𝛽)} ≰A^{c} . SinceA^{c} is an IF 𝛿closed set, we havex _{(𝛼,𝛽)} ∉A^{c} = cl_{𝛿}(A^{c} ). Then there exists an IF regular openq neighborhoodU ofx _{(𝛼,𝛽)} such that ThusU ≤A . HenceA is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}.Conversely, to show that
A^{c} is an IF 𝛿closed set, take anyx _{(𝛼,𝛽)} ∉A^{c} . Then we havex _{(𝛼,𝛽)}qA . ThusA is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}. Therefore there exists an IF regular openq neighborhoodV ofx _{(𝛼,𝛽)} such thatV ≤A^{c} , i.e.x _{(𝛼,𝛽)} ∉ cl_{𝛿}(A^{c} ). Since cl_{𝛿}(A^{c} ) ≤A^{c} , we haveA^{c} is an IF 𝛿closed set. HenceA is an IF 𝛿open set.Recall that a function
f : (X , 𝛵) → (Y, 𝛵') is said to be afuzzy 𝛿continuous function if for each fuzzy pointx _{𝛼} inX and for any fuzzy regular openq neighborhoodV off (x _{𝛼}), there exists an fuzzy regular openq neighborhoodU ofx _{𝛼} such thatf (U ) ≤V (See [18]). We define a similar definition in the intuitionistic fuzzy topological spaces as follows.Definition 2.3. A functionf : (X , 𝛵) → (Y , 𝛵') is said to beintuitionistic fuzzy 𝛿continuous if for each intuitionistic fuzzy pointx _{(𝛼,𝛽)} inX and for any intuitionistic fuzzy regular openq neighborhoodV off (x _{(𝛼,𝛽)}), there exists an intuitionistic fuzzy regular openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (U ) ≤V .Now, we characterize the intuitionistic fuzzy 𝛿continuous function in terms of IF 𝛿closure and IF 𝛿interior.
Theorem 2.4. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF 𝛿continuous function. (2) f(cl𝛿(U)) ≤ cl𝛿(f(U)) for each IF set U in X. (3) cl𝛿(f−1(V)) ≤ f−1(cl𝛿(V)) for each IF set V in Y. (4) f−1(int𝛿(V)) ≤ int𝛿(f−1(V)) for each IF set V in Y.
Proof. (1) ⇒ (2). Letx _{(𝛼,𝛽)} ∈ cl_{𝛿}(U ), and letB be an IF regular openq neighborhood off (x _{(𝛼,𝛽)}) inY . By (1), there exists an IF regular openq neighborhoodA ofx _{(𝛼,𝛽)} such thatf (A ) ≤B . Sincex _{(𝛼,𝛽)} ∈ cl_{𝛿}(U ) andA is an IF regular openq neighborhood ofx _{(𝛼,𝛽)},AqU . Sof (A )qf (U ). Sincef (A ) ≤B ,Bqf (U ). Thenf (x _{(𝛼,𝛽)}) ∈ cl_{𝛿}(f (U )). Hencef (cl_{𝛿}(U ))) ≤ cl_{𝛿}(f (U )).(2) ⇒ (3). Let
V be an IF set inY . Thenf ^{−1}(V ) is an IF set inX . 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 inX , and letV be an IF regular openq neighborhood off (x _{(𝛼,𝛽)}) inY . SinceV^{c} is an IF regular closed set,V^{c} is an IF semiopen set. By Theorem 1.3, cl(V^{c} ) = cl_{𝛿}(V^{c} ). Sincef (x _{(𝛼,𝛽)})qV ,f (x _{(𝛼,𝛽)}) ∉V^{c} = cl(V^{c} ) = cl_{𝛿}(V^{c} ). Thereforex _{(𝛼,𝛽)} ∉f ^{−1}(cl_{𝛿}(Vc )). By (3),x _{(𝛼,𝛽)} ∉ cl_{𝛿}(f ^{−1}(V^{c} )). Then there exists an IF regular openq neighborhoodU ofx _{(𝛼,𝛽)} such that SoU ≤f ^{−1}(V ), i.e.f (U ) ≤V . Hencef is an IF 𝛿continuous function.(3) ⇒ (4). Let
V be an IF set inY . By (3), cl_{𝛿}(f ^{−1}(V^{c} )) ≤f ^{−1}(cl_{𝛿}(V^{c} )). Thusf ^{−1}(int_{𝛿}(V )) =f ^{−1}((cl_{𝛿}(V^{c} ))^{c}) = (f ^{−1}(cl_{𝛿}((V^{c} ))))^{c} ≤ (cl_{𝛿}(f ^{−1}(V^{c} )))^{c} = (cl_{𝛿}((f ^{−1}(V ))^{c}))^{c} = int_{𝛿}(f ^{−1}(V )).(4) ⇒ (3). Let
V be an IF set inY . ThenV^{c} is an IF set inY . By the hypothesis,f ^{−1}(int_{𝛿}(V^{c} )) ≤ int_{𝛿}(f ^{−1}(V^{c} )). Thuscl_{𝛿}(
f ^{−1}(V )) = (int_{𝛿}((f ^{−1}(V ))^{c}))^{c} = (int_{𝛿}(f ^{−1}(V^{c} )))^{c} ≤ (f ^{−1}(int_{𝛿}(V^{c} )))^{c} =f ^{−1}((int_{𝛿}(V^{c} ))^{c}) =f ^{−1}(cl_{𝛿}(V )).Hence cl_{𝛿}(
f ^{−1}(V )) ≤f ^{−1}(cl_{𝛿}(V )).The intuitionistic fuzzy 𝛿continuous function is also characterized in terms of IF 𝛿open and IF 𝛿closed sets.
Theorem 2.5. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF 𝛿continuous function. (2) f−1(A) is an IF 𝛿closed set for each IF 𝛿closed set A in X. (3) f−1(A) is an IF 𝛿open set for each IF 𝛿open set A in X.
Proof. (1) ⇒ (2). LetA be an IF 𝛿closed set inX . ThenA = cl_{𝛿}(A ). By Theorem 2.4, cl_{𝛿}(f ^{−1}(A )) ≤f ^{−1}(cl_{𝛿}(A )) =f ^{−1}(A ). Hencef ^{−1}(A ) = cl_{𝛿}(f ^{−1}(A )). Therefore,f ^{−1}(A ) is an IF 𝛿closed set.(2) ⇒ (3). Trivial.
(3) ⇒ (1). Let
x _{(𝛼,𝛽)} be an IF point inX , and letV be an IF regular openq neighborhood off (x _{(𝛼,𝛽)}). By Corollary 1.2,V is an IF 𝛿open set. By the hypothesis,f ^{−1}(V ) is an IF 𝛿open set. Sincex _{(𝛼,𝛽)}qf ^{−1}(V ), by Lemma 2.2, we have thatf ^{−1}(V ) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}. Therefore, there exists an IF regular openq neighborhoodU ofx _{(𝛼,𝛽)} such thatU ≤f ^{−1}(V ). Hencef (U ) ≤V .The intuitionistic fuzzy 𝛿continuous function is also characterized in terms of IF 𝛿neighborhoods.
Theorem 2.6. A functionf : (X , 𝛵) → (Y , 𝛵') is IF 𝛿continuous if and only if for each IF pointx _{(𝛼,𝛽)} ofX and each IF 𝛿neighborhoodN off (x _{(𝛼,𝛽)}), the IF setf ^{−1}(N ) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}.Proof . Letx _{(𝛼,𝛽)} be an IF point inX , and letN be an IF 𝛿neighborhood off (x _{(𝛼,𝛽)}). Then there exists an IF regular openq neighborhoodV off (x _{(𝛼,𝛽)}) such thatV ≤N . Sincef is an an IF 𝛿continuous function, there exists an IF regular openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (U ) ≤V . Thus,U ≤f ^{−1}(V ) ≤N . Hencef ^{−1}(N ) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}.Conversely, let
x _{(𝛼,𝛽)} be an IF point inX , andV an IF regular openq neighborhood off (x _{(𝛼,𝛽)}). ThenV is an IF 𝛿neighborhood off (x _{(𝛼,𝛽)}). By the hypothesis,f ^{−1}(V ) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}. By the definition of IF 𝛿 neighborhood, there exists an IF regular openq neighborhoodU ofx _{(𝛼,𝛽)} such thatU ≤f ^{−1}(V ). Thusf (U ) ≤V . Hencef is an IF 𝛿continuous function.Theorem 2.7. Letf : (X , 𝛵) → (Y , 𝛵') be a bijection. Then the following statements are equivalent:(1) f is an IF 𝛿continuous function. (2) int𝛿(f(U)) ≤ f(int𝛿(U)) for each IF set U in X.
Proof. (1) ⇒ (2). LetU be an IF set inX . Thenf (U ) is an IF set inY . By Theorem 2.4,f ^{−1}(int_{𝛿}(f (U ))) ≤ int_{𝛿}(f ^{−1}(f (U ))). Sincef is onetoone,f ^{−1}(int_{𝛿}(f (U ))) ≤ int_{𝛿}(f ^{−1}(f (U ))) = int_{𝛿}(U ).Since
f is onto,int_{𝛿}(
f (U )) =f (f ^{−1}(int(f (U )))) ≤f (int(U )).(2) ⇒ (1). Let
V be an IF set inY . Thenf ^{−1}(V ) is an IF set inX . By the hypothesis, int_{𝛿} (f (f ^{−1}(V ))) ≤f (int_{𝛿}(f ^{−1}(V ))). Sincef is onto,int_{𝛿}(
V ) = int_{𝛿}(f (f ^{−1}(V ))) ≤f (int_{𝛿}(f ^{−1}(V ))).Since
f is onetoone,f ^{−1}(int_{𝛿}(V )) ≤f ^{−1}(f (int_{𝛿}(f ^{−1}(V )))) = int_{𝛿}(f ^{−1}(V )).Hence by Theorem 2.4,
f is an IF 𝛿continuous function.Recall that a function
f : (X , 𝛵) → (Y , 𝛵') is said to befuzzy weakly 𝛿continuous if for each fuzzy pointx _{𝛼} inX and each fuzzy openq neighborhoodV off (x _{𝛼}), there exists an fuzzy openq neighborhoodU ofx _{𝛼} such thatf (int(cl(U ))) ≤ cl(V ) (See [14]). We define a similar definition in the intuitionistic fuzzy topological spaces as follows.Definition 2.8. A functionf : (X , 𝛵) → (Y , 𝛵') is said to beintuitionistic fuzzy weakly 𝛿continuous if for each intuitionistic fuzzy pointx _{(𝛼,𝛽)} inX and each intuitionistic fuzzy openq neighborhoodV off (x _{(𝛼,𝛽)}), there exists an intuitionistic fuzzy openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (int(cl(U ))) ≤ cl(V ).Theorem 2.9. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF weakly 𝛿continuous function. (2) f(cl𝛿(A)) ≤ cl𝜃(f(A)) for each IF set A in X. (3) cl𝛿(f−1(B)) ≤ f−1(cl𝜃(B)) for each IF set B in Y. (4) f−1(int𝜃(B)) ≤ int𝛿(f−1(B)) for each IF set B in Y.
Proof . (1) ⇒ (2). Letx _{(𝛼,𝛽)} ∈ cl_{𝛿}(A ), and letV be an IF openq neighborhood off (x _{(𝛼,𝛽)}) inY . Sincef is an IF weakly 𝛿continuous function, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (int(cl(U ))) ≤ cl(V ). Since int(cl(V )) is an IF regular openq neighborhood ofx _{(𝛼,𝛽)} andx _{(𝛼,𝛽)} ∈ cl_{𝛿}(A ), we haveAq int(cl(V )). Thusf (A )qf (int(cl(V ))). Sincef (int(cl(V ))) ≤ cl(V ), we havef (A )q cl(V ). Thusf (x _{(𝛼,𝛽)}) ∈ cl_{𝜃}(f (A )). Hencef (cl_{𝛿}(A )) ≤ cl_{𝜃}(f (A )).(2) ⇒ (3). Let
B be an IF set inY . Thenf ^{−1}(B ) is an IF set inX . 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 inX , and letV be an IF openq neighborhood off (x _{(𝛼,𝛽)}) inY . Since cl(V ) ≤ cl(V ), Thusf (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 openq neighborhoodU ofx _{(𝛼,𝛽)} such that Thus int(cl(U )) ≤f ^{−1}(cl(V )). Therefore, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (int(cl(U ))) ≤ cl(V ). Hencef is an IF weakly 𝛿continuous function.(3) ⇒ (4). Let
B be an IF set inY . ThenB^{c} is an IF set inY . By (3), cl_{𝛿}(f ^{−1}(B^{c} )) ≤f ^{−1}(cl_{𝜃}(B^{c} )). Hence we have int_{𝛿}(f ^{−1}(B )) = (cl_{𝛿}(f ^{−1}(B^{c} ))) ≥ (f ^{−1}(cl_{𝜃}(B^{c} )))^{c} = int_{𝜃}(f ^{−1}(B )).(4) ⇒ (3). Similarly.
Theorem 2.10. A functionf : (X , 𝛵) → (Y , 𝛵') is IF weakly 𝛿continuous if and only if for each IF pointx _{(𝛼,𝛽)} inX and each IF 𝜃neighborhoodN off (x _{(𝛼,𝛽)}), the IF setf ^{−1}(N ) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}.Proof . Letx _{(𝛼,𝛽)} be an IF point inX , and letN be an IF 𝜃neighborhood off (x _{(𝛼,𝛽)}) inY . Then there exists an IF openq neighborhoodV off (x _{(𝛼,𝛽)}) such that cl(V ) ≤N . Sincef is an IF weakly 𝛿continuous function, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (int(cl(U )) ≤ cl(V ). Since cl(V ) ≤N , int(cl(U )) ≤f ^{−1}(N ). Hencef ^{−1}(N ) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}.Conversely, let
x _{(𝛼,𝛽)} be an IF point inX and letV be an IF openq neighborhood off (x _{(𝛼,𝛽)}). Since cl(V ) ≤ cl(V ), cl(V ) is an IF 𝜃neighborhood off (x _{(𝛼,𝛽)}). By the hypothesis,f ^{−1}(cl(V )) is an IF 𝛿neighborhood ofx _{(𝛼,𝛽)}. Then there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such that int(cl(V )) ≤f ^{−1}(cl(V )). Thus int(cl(V )) ≤f ^{−1}(cl(V )). Hencef is IF almost strongly 𝛿continuous.Theorem 2.11. Letf : (X , 𝛵) → (Y , 𝛵') be an IF weakly 𝛿continuous function. Then the following statements are true:(1) f−1(V) is an IF 𝜃closed set in X for each IF 𝛿closed set V in Y. (2) f−1(V) is an IF 𝜃open set in X for each IF 𝛿open set V in Y.
Proof . (1) LetB be an IF 𝜃closed set inY . Then cl_{𝜃}(B ) =B . Sincef is an IF weakly 𝛿continuous function, by Theorem 2.9, cl_{𝛿}(f ^{−1}(B )) ≤f ^{−1}(cl_{𝜃}(B )) =f ^{−1}(B ). Hencef ^{−1}(B ) is an IF 𝛿closed set inX .(2) Trivial.
Theorem 2.12. Letf : (X , 𝛵) → (Y , 𝛵') be a bijection. Then the following statements are equivalent:(1) f is an IF weakly 𝛿continuous function. (2) int𝜃(f(A)) ≤ f(int𝛿(A)) for each IF set A in X.
Proof . (1) ⇒ (2). LetA be an IF set inX . Thenf (A ) is an IF set inY . By Theorem 2.9(4),f ^{−1}(int_{𝜃}(f (A ))) ≤ int_{𝛿}(f ^{−1}(f (A ))). Since f is onetoone,f ^{−1}(int_{𝜃}(f (A ))) ≤ int_{𝛿}(f ^{−1}(f (A ))) = int_{𝛿}(A ).Since
f is onto,int_{𝜃}(
f (A )) =f (f ^{−1}(int_{𝜃}(f (A )))) ≤f (int_{𝛿}(A )).Hence int_{𝜃}(
f (A )) ≤f (int_{𝛿}(A )).(2) ⇒ (1). Let
B be an IF set inY . Thenf ^{−1}(B ) is an IF set inX . By (2) int_{𝜃}(f (f ^{−1}(B ))) ≤f (int_{𝛿}(f ^{−1}(B ))). Sincef is onto,int_{𝜃}(
B ) = int_{𝜃}(f (f ^{−1}(B )) ≤f (int_{𝛿}(f ^{−1}(B )).f is onetoone,f ^{−1}(int_{𝜃}(B ) ≤f ^{−1}(f (int_{𝛿}(f ^{−1}(B ))) = int_{𝛿}(f ^{−1}(B )).By Theorem 2.9,
f is an IF weakly 𝛿continuous function.3. IF Almost Continuous and Almost Strongly ？？continuous Functions
Definition 3.1 ( [7]). A functionf : (X , 𝛵) → (Y , 𝛵') is said to beintuitionistic fuzzy almost continuous if for any intuitionistic fuzzy regular open setV inY ,f ^{−1}(V ) is an intuitionistic fuzzy open set inX .Theorem 3.2 ( [12]). A functionf : (X , 𝛵) → (Y , 𝛵') is IF almost continuous if and only if for each IF pointx _{(𝛼,𝛽)} inX and for any IF openq neighborhoodV off (x _{(𝛼,𝛽)}), there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (U ) ≤ int(cl(V )).Theorem 3.3. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF almost continuous function. (2) f(cl(U)) ≤ cl𝛿(f(U)) for each IF set U in X. (3) f−1(V) is an IF closed set in X for each IF 𝛿closed set V in Y. (4) f−1(V) is an IF open set in X for each IF 𝛿open set V in Y.
Proof .(1) ⇒ (2). Let
x _{(𝛼,𝛽)} ∈ cl(U ). Suppose thatf (x _{(𝛼,𝛽)}) ∉ cl_{𝛿}(f (U )). Then there exists an IF openq neighborhoodV off (x _{(𝛼,𝛽)}) such that Sincef is an IF almost continuous function,f ^{−1}(V ) is an IF open set inX . SinceV qf (x _{(𝛼,𝛽)}), we havef ^{−1}(V )qx _{(𝛼,𝛽)}. Thusf ^{−1}(V ) is an IF openq neighborhood ofx _{(𝛼,𝛽)}. Sincex _{(𝛼,𝛽)} ∈ cl(U ), by Theorem 1.5, we havef ^{−1}(V )qU . Thusf (f ^{−1}(V ))qf (U ). Sincef (f ^{−1}(V )) ≤V , we haveV qf (U ). This is a contradiction. Hencef (cl(U )) ≤ cl_{𝛿}(f (U )).(2) ⇒ (3). Let
V be an IF 𝛿closed set inY . Thenf ^{−1}(V ) is an IF set inX . By the hypothesis,f (cl(f ^{−1}(V )))) ≤ cl_{𝛿}(f (f ^{−1}(V ))) ≤ cl_{𝛿}(V ) =V .Thus cl(
f ^{−1}(V )) ≤f ^{−1}(V ). Hencef ^{−1}(V ) is an IF closed set inX .(3) ⇒ (4). Let
V be an IF 𝛿open set inY . ThenV^{c} is an IF 𝛿closed set inY . By the hypothesis,f ^{−1}(V^{c} ) = (f ^{−1}(V ))^{c} is an IF closed set inX . Hencef ^{−1}(V ) is an IF open set inX .(4) ⇒ (1). Let
x _{(𝛼,𝛽)} be an IF point inX , and letV be an IF openq neighborhood off (x _{(𝛼,𝛽)}) inY . Then int(cl(V )) is an IF regular openq neighborhoodf (x _{(𝛼,𝛽)}). By Theorem 1.2, int(cl(V )) is an IF 𝛿open set inY . By the hypothesis,f ^{−1}(int(cl(V ))) is IF open inX . Since int(cl(V ))qf (x _{(𝛼,𝛽)}), we havex _{(𝛼,𝛽)}qf ^{−1}(int(cl(V ))). Thusx _{(𝛼,𝛽)} does not belong to the set (f ^{−1}(int(cl(V ))))^{c}. PutB = (f ^{−1}(int(cl(V ))))^{c}. SinceB is an IF closed set andx _{(𝛼,𝛽)} ∉B = cl(B ), there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such that Thenx _{(𝛼,𝛽)}qU ≤B^{c} =f ^{−1}(int(cl(V ))). Thusf (U ) ≤ int(cl(V )). Hence,f is an IF almost continuous function.Theorem 3.4. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF almost continuous function. (2) cl(f−1(V)) ≤ f−1(cl𝛿(V)) for each IF set V in Y. (3) int𝛿(f−1(V)) ≤ f−1(int(V)) for each IF set V in Y.
Proof . (1) ⇒ (2). LetV be an IF set inY . Thenf ^{−1}(V ) is an IF set inX . By Theorem 3.3,f (cl(f ^{−1}(V ))) ≤ cl_{𝛿}(f (f ^{−1}(V ))) ≤ cl_{𝛿}(V ).Thus cl(
f ^{−1}(V )) ≤f ^{−1}(cl_{𝛿}(V )).(2) ⇒ (1). Let
U be an IF set inX . Thenf (U ) is an IF set inY . By the hypothesis, cl(f ^{−1}(f (U ))) ≤f ^{−1}(cl_{𝛿}(f (U ))). Thencl(
U ) ≤ cl(f ^{−1}(f (U ))) ≤f ^{−1}(cl_{𝛿}(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 inY . ThenV^{c} is an IF set inY . By the hypothesis, cl(f ^{−1}(V^{c} )) ≤f ^{−1}(cl_{𝛿}(V^{c} )). Thusf ^{−1}(int_{𝛿}(V )) =f ^{−1}((cl_{𝛿}(V^{c} ))^{c}) = (f ^{−1}(cl_{𝛿}((V^{c} ))))^{c} ≤ (cl(f ^{−1}(V^{c} )))^{c} = (cl((f ^{−1}(V ))^{c} ))^{c} = int(f ^{−1}(V )).(3) ⇒ (2). Let
V be an IF set inY . ThenV^{c} is an IF set inY . By the hypothesis,f ^{−1}(int_{𝛿}(V^{c} )) ≤ int(f ^{−1}(V^{c} )). Thuscl(
f ^{−1}(V )) = (int((f ^{−1}(V ))^{c} ))^{c} = (int(f ^{−1}(V^{c} )))^{c} ≤ (f ^{−1}(int_{𝛿}(V^{c} )))^{c} =f ^{−1}((int_{𝛿}(V^{c} ))^{c} ) =f ^{−1}(cl_{𝛿}(V )).Hence cl(
f ^{−1}(V )) ≤f ^{−1}(cl_{𝛿}(V )) .Corollary 3.5. A functionf : (X , 𝛵) → (Y , 𝛵') is IF almost continuous if and only if for each IF pointx _{(𝛼,𝛽)} inX and each IF 𝛿neighborhoodN off (x _{(𝛼,𝛽)}), the IF setf ^{−1}(N ) is an IFq neighborhood ofx _{(𝛼,𝛽)}.Proof . Letx _{(𝛼,𝛽)} be an IF point inX , and letN be an IF 𝛿neighborhood off (x _{(𝛼,𝛽)}). Then there exists an IF regular openq neighborhoodV off (x _{(𝛼,𝛽)}) such thatV ≤N . Sincef is an IF almost continuous function, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (U ) ≤ int(cl(V )) =V ≤N . Thus there exists an IF open setU such thatx _{(𝛼,𝛽)}qU ≤f ^{−1}(N ). Hencef ^{−1}(N ) is an IFq neighborhood ofx _{(𝛼,𝛽)}.Conversely, let
x _{(𝛼,𝛽)} be an IF point inX , and letV be an IFq neighborhood off (x _{(𝛼,𝛽)}). Then int(cl(V )) is an IF regular openq neighborhood off (x _{(𝛼,𝛽)}). Also, int(cl(V )) is an IF 𝛿neighborhood off (x _{(𝛼,𝛽)}). By the hypothesis,f ^{−1}(int(cl(V ))) is an IFq neighborhood ofx _{(𝛼,𝛽)}. Sincef ^{−1}(int(cl(V ))) is an IFq neighborhood ofx _{(𝛼,𝛽)}, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatU ≤f ^{−1}(int(cl(V ))). Thus there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (U ) ≤ int(cl(V )). Hencef is an IF almost continuous function.Theorem 3.6. Letf : (X , 𝛵) → (Y , 𝛵') be a bijection. Then the following statements are equivalent:(1) f is an IF almost continuous function. (2) f(int𝛿(U)) ≤ int(f(U)) for each IF set U in X.
Proof . Trivial by Theorem 3.4.Recall that a function
f : (X , 𝛵) → (Y , 𝛵') is said to be afuzzy almost strongly 𝜃continuous function if for each fuzzy pointx _{𝛼} inX and each fuzzy openq neighborhoodV off (x _{𝛼}), there exists an fuzzy openq neighborhoodU ofx _{𝛼} such thatf (cl(U )) ≤ int(cl(V )) (See [14]).Definition 3.7. A functionf : (X , 𝛵) → (Y , 𝛵') is said to beintuitionistic fuzzy almost strongly 𝜃continuous if for each intuitionistic fuzzy pointx _{(𝛼,𝛽)} inX and each intuitionistic fuzzy openq neighborhoodV off (x _{(𝛼,𝛽)}), there exists an intuitionistic fuzzy openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (cl(U )) ≤ int(cl(V )).Theorem 3.8. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF almost strongly 𝜃continuous function. (2) f(cl𝜃(A)) ≤ cl𝛿(f(A)) for each IF set A in X. (3) cl𝜃(f−1(B)) ≤ f−1(cl𝛿(B)) for each IF set B in Y. (4) f−1(int𝜃(B)) ≤ int𝜃(f−1(B)) for each IF set B in Y.
Proof . (1) ⇒ (2). Letx _{(𝛼,𝛽)} ∈ cl_{𝜃}(A ). Supposef (x _{(𝛼,𝛽)}) ∉ cl_{𝛿}(f (A )). Then there exists an IF openq neighborhoodV off (x _{(𝛼,𝛽)}) such that Sincef is an IF almost strongly 𝜃 continuous function, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (cl(U )) ≤ int(cl(V )) =V . Sincef (A ) ≤V^{c} ≤ (f (cl(U )))^{c} , we haveA ≤ (f ^{−1}(f (cl(U ))))^{c} . Thus Also, Since cl(U ) ≤f ^{−1}(f (cl(U ))), we have Sincex _{(𝛼,𝛽)} ∈ cl_{𝜃}(A ), we haveAq cl(U ). This is a contradiction.(2) ⇒ (3). Let
B be an IF set inY . Thenf ^{−1}(B ) is an IF set inX . By (2),f (cl_{𝜃}(f ^{−1}(B ))) ≤ cl_{𝜃}(f (f ^{−1}(B ))) ≤ cl_{𝜃}(B ). Thus we havef (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 inY . ThenB^{c} is an IF set inY . By (3), cl_{𝜃}(f ^{−1}(B^{c} )) ≤f ^{−1}(cl_{𝛿}(B^{c} )) for each IF setB inY . Thereforef ^{−1}(int_{𝛿}(B )) = (cl_{𝜃}(f ^{−1}(B^{c} )))^{c} ≥ (f ^{−1}(cl_{𝛿}(B^{c} )))^{c} = int_{𝜃}(f ^{−1}(B )).(4) ⇒ (1). Let
B be an IF set inY . ThenB^{c} is an IF set inY . By (4),f ^{−1}(int_{𝛿}(B^{c} )) ≤ int_{𝜃}(f ^{−1}(B^{c} )). Thus cl_{𝜃}(f ^{−1}(B^{c} )) ≤f ^{−1}(cl_{𝛿}(B^{c} )). Hencef is an IF almost strongly 𝜃continuous function.Theorem 3.9. Letf : (X , 𝛵) → (Y , 𝛵') be a function. Then the following statements are equivalent:(1) f is an IF almost strongly 𝜃continuous function. (2) The inverse image of every IF 𝛿closed set in Y is an IF 𝜃closed set in X. (3) The inverse image of every IF 𝛿open set in Y is an IF 𝜃open set in X. (4) The inverse image of every IF regular open set in Y is an IF 𝜃open set in X.
Proof . (1) ⇒ (2). LetB be an IF 𝛿closed set inY . Then cl_{𝛿}(B ) =B . Sincef is an IF almost strongly 𝜃continuous 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 ). Hencef ^{−1}(B ) is an IF 𝜃closed set inX .(2) ⇒ (3). Let
B be an IF 𝛿open set inY . ThenB^{c} is an IF 𝛿closed set inY . By (4),f ^{−1}(B^{c} ) = (f ^{−1}(B ))^{c} is an IF 𝜃closed set inX . Hencef ^{−1}(B ) is an IF 𝜃open set inX .(3) ⇒ (4). Immediate since IF regular open sets are IF 𝜃open sets.
(4) ⇒ (1). Let
x _{(𝛼,𝛽)} be an IF point inX , and letV be an IF openq neighborhood off (x _{(𝛼,𝛽)}). Then int(cl(V )) is an IF regular openq neighborhood off (x _{(𝛼,𝛽)}). By (4),f ^{−1}(int(cl(V ))) is an IF 𝜃open set inX . Thenx _{(𝛼,𝛽)} ∉ (f ^{−1}(int(cl(V ))))^{c} = cl_{𝜃}((f ^{−1}(int(cl(V ))))^{c} ).Put int(cl(
V )) =D . Supposex _{(𝛼,𝛽)} ∈ (f ^{−1}(int(cl(V ))))^{c} =f ^{−1}(D^{c} ). Thenf (x _{(𝛼,𝛽)}) ∈f (f ^{−1}(D^{c} )) =f (f ^{−1}((𝛾_{D} , 𝜇_{D} ))) =f ((f ^{−1}(𝛾_{D} ),f ^{−1}(𝜇_{D} ))) = (f (f ^{−1}(𝛾_{D} )),f (f ^{−1}(𝜇_{D} ))) ⊆ (𝛾_{D} , 𝜇_{D} ).Let
f (x _{(𝛼,𝛽)}) =y _{(𝛼0,𝛽0)}. Then 𝛼_{0} ≤ 𝛾_{D} (y ) and 𝛽_{0} ≥ 𝜇_{D} (y ). SinceV is an IF open set,V ≤ int(cl(V )) =D . Thus 𝜇_{V} ≤ 𝜇_{D} and 𝛾_{v} ≥ 𝛾_{D} . Thus 𝛼_{0} ≤ 𝛾_{V} (y ) and 𝛽_{0} ≥ 𝜇_{V} (y ). SinceV is an IF openq neighborhood off (x _{(𝛼,𝛽)}), we havef (x _{(𝛼,𝛽)})qV . Thusy _{(𝛼0,𝛽0)} ≰V^{c} = (𝛾_{V} ,𝜇_{V} ). Hence 𝛼_{0} > 𝛾_{V} (y ) and 𝛽_{0} < 𝜇_{V} (y ). This is a contradiction. Therefore there exists an IF open qneighborhoodU ofx _{(𝛼,𝛽)} such that i.e. cl(U ) ≤f ^{−1}(int(cl(V ))). Thenf (cl(U )) ≤ int(cl(V )). Hencef is an IF almost strongly 𝜃continuous function.Theorem 3.10. A functionf : (X , 𝛵) → (Y , 𝛵') is IF almost strongly 𝜃continuous if and only if for each IF pointx _{(𝛼,𝛽)} inX and each IF 𝛿neighborhoodN off (x _{(𝛼,𝛽)}), the IF setf ^{−1}(N ) is an IF 𝜃neighborhood ofx _{(𝛼,𝛽)}.Proof . Letx _{(𝛼,𝛽)} be an IF point inX , and letN be an IF 𝛿neighborhood off (x _{(𝛼,𝛽)}). Then there exists an an IF regular openq neighborhoodV off (x _{(𝛼,𝛽)}) such thatV ≤N . Thus int(cl(V )) ≤N . Sincef is an IF almost strongly 𝜃 continuous function, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such thatf (cl(U )) ≤ int(cl(V )). Thusf (cl(U )) ≤N . Therefore, there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such that cl(U ) ≤f ^{−1}(N ). Hencef ^{−1}(N ) is an IF 𝜃neighborhood ofx _{(𝛼,𝛽)}.Conversely, let
x _{(𝛼,𝛽)} be an IF point inX , and letV be an IF openq neighborhood off (x _{(𝛼,𝛽)}). Since int(cl(V )) is an IF regular openq neighborhood off (x _{(𝛼,𝛽)}) and int(cl(V )) ≤ int(cl(V )), int(cl(V )) is an IF 𝛿neighborhood off (x _{(𝛼,𝛽)}). By the hypothesis,f ^{−1}(int(cl(V ))) is an IF 𝜃neighborhood ofx _{(𝛼,𝛽)}. Then there exists an IF openq neighborhoodU ofx _{(𝛼,𝛽)} such that cl(U ) ≤f ^{−1}(int(cl(V ))). Thereforef (cl(U )) ≤ int(cl(V )). Hencef is IF almost strongly 𝜃continuous.Theorem 3.11. Letf : (X , 𝛵) → (Y , 𝛵') be a bijection. Then the following statements are equivalent:(1) f is an IF almost strongly 𝜃continuous function. (2) int𝛿(f(A)) ≤ f(int𝜃(A)) for each IF set A in X.
Proof . (1) ⇒ (2). LetA be an IF set inX . Thenf (A ) is an IF set inY . By Theorem 3.9,f ^{−1}(int_{𝛿}(f (A ))) ≤ int_{𝜃}(f ^{−1}(f (A ))). Sincef is onetoone,f ^{−1}(int_{𝛿}(f (A ))) ≤ int_{𝜃}(f ^{−1}(f (A ))) = int_{𝜃}(A ).Since
f is onto,int_{𝛿}(
f (A )) =f (f ^{−1}(int_{𝛿}(f (A )))) ≤f (int_{𝜃}(A )).(2) ⇒ (1). Let
B be an IF set inY . Thenf ^{−1}(B ) is an IF set inX . By (2), int_{𝛿}(f (f ^{−1}(B ))) ≤f (int_{𝜃}(f ^{−1}(B ))). Sincef is onto,int_{𝛿}(
B ) = int_{𝛿}(f (f ^{−1}(B ))) ≤f (int_{𝜃}(f ^{−1}(B ))).Since
f is onetoone,f ^{−1}(int_{𝛿}(B )) ≤f ^{−1}(f (int_{𝜃}(f ^{−1}(B )))) = int_{𝜃}(f ^{−1}(B )).By Theorem 3.9,
f is an IF almost strongly 𝜃continuous function.4. Conclusion
We characterized the intuitionistic fuzzy 𝛿continuous functions in terms of IF 𝛿closure and IF 𝛿interior, or IF 𝛿open and IF 𝛿closed sets, or IF 𝛿neighborhoods.
Moreover, we characterized the IF weakly 𝛿continuous, IF almost continuous, and IF almost strongly 𝜃continuous functions in terms of closure and interior.
Conflict of Interest
No potential conflict of interest relevant to this article was reported.