Intuitionistic Fuzzy ??-continuous Functions

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  • 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.


  • 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 [24] introduced the intuitionistic fuzzy topological space, which is a generalization of the fuzzy topological space. Moreover, many researchers have studied about this space [512].

    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, 𝛾A), where the functions 𝜇A : XI and 𝛾A : XI denote the degree of membership and the degree of nonmembership, 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 “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 semi-open in X. (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) = cl𝛿(A).

    Lemma 1.4 ( [17]). (1) For any IF set U 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 that x(𝛼,𝛽)qU, int(cl(U)) is an IF regular open q-neighborhood of x(𝛼,𝛽).

    Theorem 1.5 ( [12]). Let x(𝛼,𝛽) be an IF point in X, and U = (𝜇U, 𝛾U) 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 ??-continuous and Weakly ??-continuous Functions

    Recall that a fuzzy set N in (X, 𝛵) 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 or equivalently VN (See [14]). Now, we define a similar definition in the intuitionistic fuzzy topological spaces.

    Definition 2.1. An intuitionistic fuzzy set N in (X, 𝛵) 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

    VN.

    Lemma 2.2. An IF set A is an IF 𝛿-open set in (X, 𝛵) 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, 𝛵) such that x(𝛼,𝛽)qA. Then x(𝛼,𝛽) Ac. Since Ac is an IF 𝛿-closed set, we have x(𝛼,𝛽)Ac = cl𝛿(Ac). Then there exists an IF regular open q-neighborhood U of x(𝛼,𝛽) such that Thus UA. 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 VAc, i.e. x(𝛼,𝛽) ∉ cl𝛿(Ac). Since cl𝛿(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, 𝛵) → (Y, 𝛵') 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(x𝛼), there exists an fuzzy regular open q-neighborhood U of x𝛼 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, 𝛵) → (Y, 𝛵') 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

    f(U) ≤ V.

    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, 𝛵) → (Y, 𝛵') be a function. Then the following statements are equivalent:

    (1) f is an IF 𝛿-continuous 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). Let x(𝛼,𝛽) ∈ cl𝛿(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) ∈ cl𝛿(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)) ∈ cl𝛿(f(U)). Hence f(cl𝛿(U))) ≤ cl𝛿(f(U)).

    (2) ⇒ (3). Let V be an IF set in Y. Then f−1(V) is an IF set in X. By (2), f(cl𝛿(f−1(V))) ≤ cl𝛿(f(f−1(V))) ≤ cl𝛿(V). Thus cl𝛿(f−1(V)) ≤ f−1(cl𝛿(V)).

    (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)) in Y. Since Vc is an IF regular closed set, Vc is an IF semi-open set. By Theorem 1.3, cl(Vc) = cl𝛿(Vc). Since f(x(𝛼,𝛽))qV, f(x(𝛼,𝛽)) ∉ Vc = cl(Vc) = cl𝛿(Vc). Therefore x(𝛼,𝛽)f−1(cl𝛿(Vc)). By (3), x(𝛼,𝛽) ∉ cl𝛿(f−1(Vc)). Then there exists an IF regular open q-neighborhood U of x(𝛼,𝛽) such that So Uf−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), cl𝛿(f−1(Vc)) ≤ f−1(cl𝛿(Vc)). Thus

    f−1(int𝛿(V)) = f−1((cl𝛿(Vc))c) = (f−1(cl𝛿((Vc))))c ≤ (cl𝛿(f−1(Vc)))c = (cl𝛿((f−1(V))c))c = int𝛿(f−1(V)).

    (4) ⇒ (3). Let V be an IF set in Y. Then Vc is an IF set in Y. By the hypothesis, f−1(int𝛿(Vc)) ≤ int𝛿(f−1(Vc)). Thus

    cl𝛿(f−1(V)) = (int𝛿((f−1(V))c))c = (int𝛿(f−1(Vc)))c ≤ (f−1(int𝛿(Vc)))c = f−1((int𝛿(Vc))c) = f−1(cl𝛿(V)).

    Hence cl𝛿(f−1(V)) ≤ f−1(cl𝛿(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, 𝛵) → (Y, 𝛵') be a function. Then the following statements are equivalent:

    (1) f is an IF 𝛿-continuous function. (2) f−1(A) is an IF 𝛿-closed set for each IF 𝛿-closed set A in X. (3) f−1(A) is an IF 𝛿-open set for each IF 𝛿-open set A in X.

    Proof. (1) ⇒ (2). Let A be an IF 𝛿-closed set in X. Then A = cl𝛿(A). By Theorem 2.4, cl𝛿(f−1(A)) ≤ f−1(cl𝛿(A)) = f−1(A). Hence f−1(A) = cl𝛿(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 Uf−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, 𝛵) → (Y, 𝛵') 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 VN. 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, Uf−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 Uf−1(V). Thus f(U) ≤ V. Hence f is an IF 𝛿-continuous function.

    Theorem 2.7. Let f : (X, 𝛵) → (Y, 𝛵') be a bijection. Then the following statements are equivalent:

    (1) f is an IF 𝛿-continuous function. (2) int𝛿(f(U)) ≤ f(int𝛿(U)) for each IF set U in X.

    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(int𝛿(f(U))) ≤ int𝛿(f−1(f(U))). Since f is one-to-one,

    f−1(int𝛿(f(U))) ≤ int𝛿(f−1(f(U))) = int𝛿(U).

    Since f is onto,

    int𝛿(f(U)) = f(f−1(int(f(U)))) ≤ f(int(U)).

    (2) ⇒ (1). Let V be an IF set in Y. Then f−1(V) is an IF set in X. By the hypothesis, int𝛿 (f(f−1(V))) ≤ f(int𝛿(f−1(V))). Since f is onto,

    int𝛿(V) = int𝛿(f(f−1(V))) ≤ f(int𝛿(f−1(V))).

    Since f is one-to-one,

    f−1(int𝛿(V)) ≤ f−1(f(int𝛿(f−1(V)))) = int𝛿(f−1(V)).

    Hence by Theorem 2.4, f is an IF 𝛿-continuous function.

    Recall that a function f : (X, 𝛵) → (Y, 𝛵') 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, 𝛵) → (Y, 𝛵') 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

    f(int(cl(U))) ≤ cl(V).

    Theorem 2.9. Let f : (X, 𝛵) → (Y, 𝛵') be a function. Then the following statements are equivalent:

    (1) f is an IF weakly 𝛿-continuous 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). Let x(𝛼,𝛽) ∈ cl𝛿(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) ∈ cl𝛿(A), we have Aqint(cl(V)). Thus f(A)qf(int(cl(V))). Since f(int(cl(V))) ≤ cl(V), we have f(A)qcl(V). Thus f(x(𝛼,𝛽)) ∈ cl𝜃(f(A)). Hence f(cl𝛿(A)) ≤ cl𝜃(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(cl𝛿(f−1(B))) ≤ cl𝜃(f(f−1(B))) ≤ cl𝜃(B). Hence cl𝛿(f−1(B)) ≤ f−1(cl𝜃(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), Thus f(x(𝛼,𝛽)) ∉ cl𝜃((cl(V))c). By (3), f(x(𝛼,𝛽)) ∉ cl𝛿(f−1((cl(V))c)). Then there exists an intuitionistic fuzzy regular open q-neighborhood U of x(𝛼,𝛽) such that 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), cl𝛿(f−1(Bc)) ≤ f−1(cl𝜃(Bc)). Hence we have int𝛿(f−1(B)) = (cl𝛿(f−1(Bc))) ≥ (f−1(cl𝜃(Bc)))c = int𝜃(f−1(B)).

    (4) ⇒ (3). Similarly.

    Theorem 2.10. A function f : (X, 𝛵) → (Y, 𝛵') 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)) ≤ cl(V). Since cl(V) ≤ N, int(cl(U)) ≤ 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 𝛿-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, 𝛵) → (Y, 𝛵') be an IF weakly 𝛿-continuous function. Then the following statements are true:

    (1) f−1(V) is an IF 𝜃-closed set in X for each IF 𝛿-closed set V in Y. (2) f−1(V) is an IF 𝜃-open set in X for each IF 𝛿-open set V in Y.

    Proof. (1) Let B be an IF 𝜃-closed set in Y. Then cl𝜃(B) = B. Since f is an IF weakly 𝛿-continuous function, by Theorem 2.9, cl𝛿(f−1(B)) ≤ f−1(cl𝜃(B)) = f−1(B). Hence f−1(B) is an IF 𝛿-closed set in X.

    (2) Trivial.

    Theorem 2.12. Let f : (X, 𝛵) → (Y, 𝛵') be a bijection. Then the following statements are equivalent:

    (1) f is an IF weakly 𝛿-continuous function. (2) int𝜃(f(A)) ≤ f(int𝛿(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 2.9-(4), f−1(int𝜃(f(A))) ≤ int𝛿(f−1(f(A))). Since f is one-to-one,

    f−1(int𝜃(f(A))) ≤ int𝛿(f−1(f(A))) = int𝛿(A).

    Since f is onto,

    int𝜃(f(A)) = f(f−1(int𝜃(f(A)))) ≤ f(int𝛿(A)).

    Hence int𝜃(f(A)) ≤ f(int𝛿(A)).

    (2) ⇒ (1). Let B be an IF set in Y. Then f−1(B) is an IF set in X. By (2) int𝜃(f(f−1(B))) ≤ f(int𝛿(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 ??-continuous Functions

    Definition 3.1 ( [7]). A function f : (X, 𝛵) → (Y, 𝛵') 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, 𝛵) → (Y, 𝛵') 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 f(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, 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 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)) ∉ cl𝛿(f(U)). Then there exists an IF open q-neighborhood V of f(x(𝛼,𝛽)) such that Since f is an IF almost continuous function, f−1(V) is an IF open set in X. Since V qf(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 V qf(U). This is a contradiction. Hence f(cl(U)) ≤ cl𝛿(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)qf−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 Then x(𝛼,𝛽)qUBc = 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, 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). 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(cl𝛿(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(cl𝛿(f(U))). Then

    cl(U) ≤ cl(f−1(f(U))) ≤ f−1(cl𝛿(f(U))).

    Thus f(cl(U)) ≤ cl𝛿(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(cl𝛿(Vc)). Thus

    f−1(int𝛿(V)) = f−1((cl𝛿(Vc))c) = (f−1(cl𝛿((Vc))))c ≤ (cl(f−1(Vc)))c = (cl((f−1(V))c))c = int(f−1(V)).

    (3) ⇒ (2). Let V be an IF set in Y. Then Vc is an IF set in Y. By the hypothesis, f−1(int𝛿(Vc)) ≤ int(f−1(Vc)). Thus

    cl(f−1(V)) = (int((f−1(V))c))c = (int(f−1(Vc)))c ≤ (f−1(int𝛿(Vc)))c = f−1((int𝛿(Vc))c) = f−1(cl𝛿(V)).

    Hence cl(f−1(V)) ≤ f−1(cl𝛿(V)) .

    Corollary 3.5. A function f : (X, 𝛵) → (Y, 𝛵') 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 VN. Since f is an IF almost continuous function, there exists an IF open q-neighborhood U of x(𝛼,𝛽) such that f(U) ≤ int(cl(V)) = VN. Thus there exists an IF open set U such that x(𝛼,𝛽)qUf−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(𝛼,𝛽)). 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 Uf−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, 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 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, 𝛵) → (Y, 𝛵') 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, 𝛵) → (Y, 𝛵') be a function. Then the following statements are equivalent:

    (1) f is an IF almost strongly 𝜃-continuous 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). Let x(𝛼,𝛽) ∈ cl𝜃(A). Suppose f(x(𝛼,𝛽)) ∉ cl𝛿(f(A)). Then there exists an IF open q-neighborhood V of f(x(𝛼,𝛽)) such that 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 Also, Since cl(U) ≤ f−1(f(cl(U))), we have Since x(𝛼,𝛽) ∈ cl𝜃(A), we have Aqcl(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(cl𝜃(f−1(B))) ≤ cl𝜃(f(f−1(B))) ≤ cl𝜃(B). Thus we have f(cl𝜃(f−1(B))) ≤ cl𝜃(f(f−1(B))) ≤ cl𝜃(B). Hence cl𝜃(f−1(B)) ≤ f−1(cl𝛿(B)).

    (3) ⇒ (4). Let B be an IF set in Y. Then Bc is an IF set in Y. By (3), cl𝜃(f−1(Bc)) ≤ f−1(cl𝛿(Bc)) for each IF set B in Y. Therefore f−1(int𝛿(B)) = (cl𝜃(f−1(Bc)))c ≥ (f−1(cl𝛿(Bc)))c = int𝜃(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(int𝛿(Bc)) ≤ int𝜃(f−1(Bc)). Thus cl𝜃(f−1(Bc)) ≤ f−1(cl𝛿(Bc)). Hence f is an IF almost strongly 𝜃-continuous function.

    Theorem 3.9. Let f : (X, 𝛵) → (Y, 𝛵') 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 cl𝛿(B) = B. Since f is an IF almost strongly 𝜃-continuous function, by Theorem 3.8, cl𝜃(f−1(B)) ≤ f−1(cl𝛿(B)) = f−1(B). Thus cl𝜃(f−1(B)) = f−1(B). Hence f−1(B) is an IF 𝜃-closed set in X.

    (2) ⇒ (3). Let B be an IF 𝛿-open set in Y. Then Bc is an IF 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

    x(𝛼,𝛽) ∉ (f−1(int(cl(V))))c = cl𝜃((f−1(int(cl(V))))c).

    Put int(cl(V)) = D. Suppose x(𝛼,𝛽) ∈ (f−1(int(cl(V))))c = f−1(Dc). Then

    f(x(𝛼,𝛽)) ∈ f(f−1(Dc)) = f(f−1((𝛾D, 𝜇D))) = f((f−1(𝛾D), f−1(𝜇D))) = (f(f−1(𝛾D)), f(f−1(𝜇D))) ⊆ (𝛾D, 𝜇D).

    Let f(x(𝛼,𝛽)) = y(𝛼0,𝛽0). Then 𝛼0 ≤ 𝛾D(y) and 𝛽0 ≥ 𝜇D(y). Since V is an IF open set, V ≤ int(cl(V)) = D. Thus 𝜇V ≤ 𝜇D and 𝛾v ≥ 𝛾D. Thus 𝛼0 ≤ 𝛾V(y) and 𝛽0 ≥ 𝜇V(y). Since V is an IF open q-neighborhood of f(x(𝛼,𝛽)), we have f(x(𝛼,𝛽))qV. Thus y(𝛼0,𝛽0) Vc = (𝛾V,𝜇V). Hence 𝛼0 > 𝛾V(y) and 𝛽0 < 𝜇V(y). This is a contradiction. Therefore there exists an IF open q-neighborhood U of x(𝛼,𝛽) such that 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, 𝛵) → (Y, 𝛵') 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 VN. 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, 𝛵) → (Y, 𝛵') be a bijection. Then the following statements are equivalent:

    (1) f is an IF almost strongly 𝜃-continuous function. (2) int𝛿(f(A)) ≤ f(int𝜃(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(int𝛿(f(A))) ≤ int𝜃(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), int𝛿(f(f−1(B))) ≤ f(int𝜃(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 𝜃-continuous functions in terms of closure and interior.

    Conflict of Interest

    No potential conflict of interest relevant to this article was reported.

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