In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy θ -compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzy θ -compactness is strictly weaker than intuitionistic fuzzy compactness. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzy θ -compactness in the original intuitionistic fuzzy topology. This characterization shows that intuitionistic fuzzy θ -compactness can be related to an appropriated notion of intuitionistic fuzzy convergence. 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 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, 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. 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 and Theorem 2.4. [2] For any IF set A in an IF topological space ( X, Τ ), we have 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. 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 ϵ-neighborhood ( q-neighborhood ) of x _(α,β) if there exists an intuitionistic fuzzy open set U in X such that x _(α,β) ∈ U ≤ A ( x _(α,β) qU ≤ A , respectively). Theorem 2.10. [10] Let x _(α,β) and U = ( 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) , Aq cl( U ). The set of all intuitionistic fuzzy θ -cluster points of A is called intuitionistic fuzzy θ-closure of A and is denoted by cl _θ ( A ). An intuitionistic fuzzy set A is called an intuitionistic fuzzy θ-closed set if A = cl _θ ( A ). The complement of an intuitionistic fuzzy θ -closed set is said to be an intuitionistic fuzzy θ-open set . Definition 2.12. [11] Let ( X,Τ ) and U denote an intuitionistic fuzzy topological space and an intuitionistic fuzzy set in X , respectively. The intuitionistic fuzzy θ-interior of U is denoted and defined by 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. Remark 2.17. Intuitionistic fuzzy sets have some different properties compared to fuzzy sets, and these properties are shown in the subsequent examples. 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 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 )), 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: 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, 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 )), 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 I ₀ of I such that A ≤ ˅ { G_i | i ∈ I ₀ }. Remark 4.4. Since every IF θ -open set is IF open, it follows that every IF compact space is IF θ -compact. Theorem 4.5. An IF topological space ( X,Τ ) is IF θ -compact if and only if every family of IF θ -closed subsets of X with the finite intersection property has a nonempty intersection. Proof. Let X be IF θ -compact and let F = { 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 I ₀ 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 { G ₁ , G ₂ , … , G_n } in g such that This implies that { G ₁ , G ₂ , … , 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 I ₀ 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] ). 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 I ₀ of I such that Since f is onto, { G_i | i ∈ I ₀ } 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 I ₀ of I such that A ≤ ˅ { f ^–1 ( G_i ) | i ∈ I ₀ }. Therefore, f ( A ) ≤ ˅ { G_i | i ∈ I ₀ }. Hence, f ( A ) is IF θ -compact relative to Y . Theorem 4.11. Let A and B be subsets of an IF topological space ( X,Τ ). If A is IF θ -compact and B is IF θ -closed in X , then A ∧ B is IF θ -compact. Proof. Let g = { 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 I ₀ of I such that A ≤ ( ˅ { G_i | i ∈ I ₀ }) ∨ B^c . Therefore, A ∧ B ≤ ( ˅ { G_i | i ∈ I ₀ }). 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.