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 = (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} in X 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 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) = cl_{𝛿}(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 = (𝜇_{U}, 𝛾_{U}) 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)}.
Recall that a fuzzy set N in (X, 𝛵) is said to be a fuzzy 𝛿-neighborhood of a fuzzy point x_{𝛼} if there exists a fuzzy regular open q-neighborhood V of x𝛼 such that 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, 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
V ≤ N.
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)} ≰ A^{c}. Since A^{c} is an IF 𝛿-closed set, we have x_{(𝛼,𝛽)} ∉ A^{c} = cl_{𝛿}(A^{c}). Then there exists an IF regular open q-neighborhood U of x_{(𝛼,𝛽)} such that Thus U ≤ A. Hence A is an IF 𝛿-neighborhood of x_{(𝛼,𝛽)}.
Conversely, to show that A^{c} is an IF 𝛿-closed set, take any x_{(𝛼,𝛽)} ∉ A^{c}. 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 ≤ A^{c}, i.e. x_{(𝛼,𝛽)} ∉ cl_{𝛿}(A^{c}). Since cl_{𝛿}(A^{c}) ≤ A^{c}, we have A^{c} 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 x_{𝛼} 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, 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)} ∈ 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(cl_{𝛿}(U))) ≤ cl_{𝛿}(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(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 IF regular open q-neighborhood of f(x_{(𝛼,𝛽)}) in Y. Since V^{c} is an IF regular closed set, V^{c} is an IF semi-open set. By Theorem 1.3, cl(V^{c}) = cl_{𝛿}(V^{c}). Since f(x_{(𝛼,𝛽)})qV, f(x_{(𝛼,𝛽)}) ∉ V^{c} = cl(V^{c}) = cl_{𝛿}(V^{c}). Therefore x_{(𝛼,𝛽)} ∉ f^{−1}(cl_{𝛿}(Vc)). By (3), x_{(𝛼,𝛽)} ∉ cl_{𝛿}(f^{−1}(V^{c})). Then there exists an IF regular open q-neighborhood U of x_{(𝛼,𝛽)} such that 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}(V^{c})) ≤ f^{−1}(cl_{𝛿}(V^{c})). Thus
f^{−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 in Y. Then V^{c} is an IF set in Y. By the hypothesis, f^{−1}(int_{𝛿}(V^{c})) ≤ int_{𝛿}(f^{−1}(V^{c})). Thus
cl_{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 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 = 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, 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 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, 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}(int_{𝛿}(f(U))) ≤ int_{𝛿}(f^{−1}(f(U))). Since f is one-to-one,
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 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,
int_{𝛿}(V) = int_{𝛿}(f(f^{−1}(V))) ≤ f(int_{𝛿}(f^{−1}(V))).
Since f is one-to-one,
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 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 x_{𝛼} in X and each fuzzy open q-neighborhood V of f(x_{𝛼}), there exists an fuzzy open q-neighborhood U of x_{𝛼} 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)} ∈ 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 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)}) ∈ 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), 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 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 B^{c} is an IF set in Y. 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 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 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, 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}(int_{𝜃}(f(A))) ≤ int_{𝛿}(f^{−1}(f(A))). Since f is one-to-one,
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 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,
int_{𝜃}(B) = int_{𝜃}(f(f^{−1}(B)) ≤ f(int_{𝛿}(f^{−1}(B)).
f is one-to-one,
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 𝛿-continuous function.
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)}) ∉ cl_{𝛿}(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)) ≤ 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)))) ≤ cl_{𝛿}(f(f^{−1}(V))) ≤ cl_{𝛿}(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 V^{c} is an IF 𝛿-closed set in Y. By the hypothesis, f^{−1}(V^{c}) = (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)}qU ≤ B^{c} = 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))) ≤ 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 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}(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 in Y. Then V^{c} is an IF set in Y. By the hypothesis, cl(f^{−1}(V^{c})) ≤ f^{−1}(cl_{𝛿}(V^{c})). Thus
f^{−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 in Y. Then V^{c} is an IF set in Y. By the hypothesis, f^{−1}(int_{𝛿}(V^{c})) ≤ int(f^{−1}(V^{c})). Thus
cl(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 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 V ≤ N. 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)) = V ≤ N. Thus there exists an IF open set U such that x_{(𝛼,𝛽)}qU ≤ 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_{(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 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, 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 x_{𝛼} in X and each fuzzy open q-neighborhood V of f(x_{𝛼}), there exists an fuzzy open q-neighborhood U of x_{𝛼} 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)} ∈ 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)}) 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) ≤ V^{c} ≤ (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)} ∈ cl_{𝜃}(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(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 B^{c} is an IF set in Y. By (3), cl_{𝜃}(f^{−1}(B^{c})) ≤ f^{−1}(cl_{𝛿}(B^{c})) for each IF set B in Y. Therefore f^{−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 in Y. Then B^{c} is an IF set in Y. 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})). 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 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 𝜃-closed set in X.
(2) ⇒ (3). Let B be an IF 𝛿-open set in Y. Then B^{c} is an IF 𝛿-closed set in Y. By (4), f^{−1}(B^{c}) = (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} = cl_{𝜃}((f^{−1}(int(cl(V))))^{c}).
Put int(cl(V)) = D. Suppose x_{(𝛼,𝛽)} ∈ (f^{−1}(int(cl(V))))^{c} = f^{−1}(D^{c}). Then
f(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). Since V 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). Since V is an IF open q-neighborhood of f(x_{(𝛼,𝛽)}), we have f(x_{(𝛼,𝛽)})qV. Thus y_{(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 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 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, 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}(int_{𝛿}(f(A))) ≤ int_{𝜃}(f^{−1}(f(A))). Since f is one-to-one,
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 in Y. Then f^{−1}(B) is an IF set in X. By (2), int_{𝛿}(f(f^{−1}(B))) ≤ f(int_{𝜃}(f^{−1}(B))). Since f is onto,
int_{𝛿}(B) = int_{𝛿}(f(f^{−1}(B))) ≤ f(int_{𝜃}(f^{−1}(B))).
Since f is one-to-one,
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 𝜃-continuous function.
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.
No potential conflict of interest relevant to this article was reported.