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 I_{0} of Isuch 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]).
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 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.