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.

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 μ_A : X → I and γ_A : X → I denote the degree of membership and the degree of non-membership, 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 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 = (μ_A, γ_A), B = (μ_B, γ_B). Then,

(1) A ≤ B iff μA(x) ≤ μB(x) and γA(x) ≥ γB(x) for all x ∈ X,

(2) A = B iff A ≤ B and B ≤ A,

(3) Ac = (γA,μA),

(4) A ∩ B = (μA ∧ μB, γA ∨ γB),

(5) A ∪ B = (μA ∨ μB, γA ∧ γB),

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.

(2) G1 ∩ G2 ∈ Τ for any G1, G2 ∈ Τ,

(3) ？Gi ∈ Τ for 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))c and 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

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 = (μ_A, γ_A) in X, denoted by x_(α,β) ∈ A, if α ≤ μ_A(x) and β ≥ γ_A(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 point x(α,β) is said to be quasicoincident with the intuitionistic fuzzy set U = (μU,γU), denoted by x(α,β)qU, if α > γU(x) or β < μU(x).

(2) Let U = (μU,γU) and V = (μV,γV) denote two intuitionistic fuzzy sets in X. Then, U and V are said to be quasi-coincident, denoted by UqV, if there exists an element x ∈ X such that μU(x) > γV(x) or γU(x) < μV(x).

The word ‘not quasi-coincident’ will be abbreviated as

herein.

Proposition 2.8. [4] Let U, V, and x_(α,β) denote IF sets and an IF point in X, respectively. Then,

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 ？-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 = (μ_U,γ_U) 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), Aqcl(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) If f(x(α,β))qV, then x(α,β)qf？1(V) for any IF set V in Y.

(2) If x(α,β)qU, then f(x(α,β))qf(U) for any IF set U in X.

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) ∈ A or x(α,β) ∈ B.

2. x(α,β)qA and x(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

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

In addition, let x = ¼, (α,β) = (½,¼). Then, x_(α,β)qA and x_(α,β)qB; however,

For the notions that are not mentioned in this section, refer to [11].

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 Τ_θ be an IF topology on X generated using the subbase of all the IF θ-open sets in X, and let U_θ 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,Τ_θ) → (Y,U_θ) is IF continuous.

Proof. Trivial.

Recall that a fuzzy set A is said to be a fuzzy θ-neighborhood of a fuzzy point x_α if there exists a fuzzy closed q-neighborhood U of x_α, such that

[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 x_α in X and each fuzzy open q-neighborhood V of f(x_α), there exists a fuzzy open q-neighborhood U of x_α 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 set A in X,

(2) f(cl(int(cl(f？1(B))))) ≤ clθ(B) for each IF set B in Y .

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) f is an IF weakly θ-continuous function.

(2) For each IF open set U with x(α,β)qf？1(U), x(α,β)q int(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.

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.

Definition 4.1. A collection {G_i | 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 ≤ ？{G_i | 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 {G_i | i ∈ I} of intuitionistic fuzzy θ-open sets of X such that A ≤ ？{G_i | i ∈ I}, there is a finite subset I₀ of Isuch that A ≤ ？{G_i | i ∈ I₀}.

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 = {F_i | i ∈ I} denote any family of IF θ-closed subsets of X with the finite intersection property. Suppose that

Then,

is an IF θ-open cover of X. Since X is IF θ-compact, there is a finite subset I₀ of I such that

This implies that

which contradicts the assumption that F has a finite intersection property. Hence,

Let g = {G_i | i ∈ I} denote an IF θ-open cover of X and consider the family

of an IF θ-closed set. Since g is a cover of X,

Hence, g' does not have the finite intersection property, i.e., there are finite numbers of IF θ-open sets {G₁, G₂, … , G_n} in g such that

This implies that {G₁, G₂, … , G_n} 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 Τ_θ 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,Τ_θ) is IF compact.

Proof. Let (X,Τ_θ) be IF compact and let g = {G_i | i ∈ I} denote an IF θ-open cover of X in T. Since for each i ∈ I, G_i ∈ Τ_θ, g is an IF open cover of X in Τ_θ. Since (X,Τ_θ) 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 = {G_i | G_i ∈ Τ_θ, i ∈ I} denote an IF open cover of X in Τ_θ. Since for each i ∈ I, G_i ∈ Τ_θ, G_i 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,Τ_θ) 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 = {G_i | i ∈ I} denote an IF θ-open cover of A. Since A^c is an IF θ-open subset of X, g = {G_i | i ∈ I} ∪ A^c is an IF θ-open cover of X. Since X is IF θ-compact, there is a finite subset I₀ of I such that

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 Τ_θ 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 Τ_θ denote an IF topology on X generated by the subbase of all IF θ-open sets in X and let U_θ 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,Τ_θ) → (Y,U_θ) 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 strongly θ-continuous image of an IF θ-compact set is IF compact.

(2) Let (X,Τ) and (Y,U) denote IF topological spaces and let f : (X,Τ) → (Y,U) be IF θ-irresolute. If a subset A of X is IF θ-compact, then image f(A) is IF θ-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 = {G_i | i ∈ I} be an IF open cover of Y. Since f is an IF strongly θ-continuous function, f : (X,Τ_θ) → (Y,U) is an IF continuous function (Theorem 4.2 of [11]). Therefore, {f^？1(G_i) | i ∈ I} is an IF θ-open cover of X. Since X is IF θ-compact, there is a finite subset I₀ of I such that

Since f is onto, {G_i | i ∈ I₀} is a finite subcover of Y. Hence, Y is IF compact.

(2) Let g = {G_i | i ∈ I} be an IF θ-open cover of f(A) in Y. Since f is an IF θ-irresolute, for each G_i, f^？1(G_i) is an IF θ-open set. Moreover, {f^？1(G_i) | i ∈ I} is an IF θ-open cover of A. Since A is IF θ-compact relative to X, there exists a finite subset I₀ of I such that A ≤ ？{f^？1(G_i) | i ∈ I₀}. Therefore, f(A) ≤ ？{G_i | i ∈ I₀}. 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 = {G_i | i ∈ I} be an IF θ-open cover of A ？ B in X. Since B^c is IF θ-open in X, (？{G_i | i ∈ I}) ∨ B^c is an IF θ-open cover of A. Since A is IF θ-compact, there is a finite subset I₀ of I such that A ≤ (？{G_i | i ∈ I₀}) ∨ B^c. Therefore, A∧B ≤ (？{G_i | i ∈ I₀}). Hence, A∧B is IF θ-compact.

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.

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