Intuitionistic Fuzzy Theta-Compact Spaces

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  • ABSTRACT

    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.


  • KEYWORD

    Intuitionistic fuzzy topology , Theta-compact

  • 1. Introduction

    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.

    2. Preliminaries

    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 μA : XI and γA : XI denote the degree of membership and the degree of non-membership, 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 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 = (μA, γA), B = (μB, γB). 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),

    image

    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.

    image

    (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

    image

    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 = (μA, γA) in X, denoted by x(α,β)A, if αμA(x) and βγA(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

    image

    herein.

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

    image
    image
    image
    image

    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)UA (x(α,β)qUA, respectively).

    Theorem 2.10. [10] Let x(α,β) and U = (μU,γU) 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 clθ(A). An intuitionistic fuzzy set A is called an intuitionistic fuzzy θ-closed set if A = clθ(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 : XY 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 : XY 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

    image

    In addition, let x = ¼, (α,β) = (¼,½). Then, x(α,β)AB. 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

    image

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

    image

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

    3. Intuitionistic Fuzzy θ-Irresolute and Weakly θ-Continuity

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

    Proof. Trivial.

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

    image

    [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 xα in X and each fuzzy open q-neighborhood V of f(xα), there exists a fuzzy open q-neighborhood U of xα 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 Uf?1(N). Therefore, there exists an IF q-neighborhood U of x(α,β) such that Uf?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)qUf?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)) ∈ clθ(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)) ∈ clθ(B). Hence we obtain f(cl(int(cl(f?1(B))))) ≤ clθ(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 Vf?1(cl(U)), i.e.

    image

    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.

    4. Intuitionistic Fuzzy θ-Compactness

    Definition 4.1. A collection {Gi | iI} 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{Gi | iI}.

    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 {Gi | iI} of intuitionistic fuzzy θ-open sets of X such that A{Gi | iI}, there is a finite subset I0 of Isuch that A{Gi | iI0}.

    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 = {Fi | iI} denote any family of IF θ-closed subsets of X with the finite intersection property. Suppose that

    image

    Then,

    image

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

    image

    This implies that

    image

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

    image

    Let g = {Gi | iI} denote an IF θ-open cover of X and consider the family

    image

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

    image

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

    image

    This implies that {G1, G2, … , Gn} 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 Τθ 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,Τθ) is IF compact.

    Proof. Let (X,Τθ) be IF compact and let g = {Gi | iI} denote an IF θ-open cover of X in T. Since for each iI, GiΤθ, g is an IF open cover of X in Τθ. Since (X,Τθ) 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 = {Gi | GiΤθ, iI} denote an IF open cover of X in Τθ. Since for each iI, GiΤθ, Gi 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,Τθ) 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 = {Gi | iI} denote an IF θ-open cover of A. Since Ac is an IF θ-open subset of X, g = {Gi | iI} ∪ Ac is an IF θ-open cover of X. Since X is IF θ-compact, there is a finite subset I0 of I such that

    image

    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 Τθ 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 Τθ denote an IF topology on X generated by the subbase of all IF θ-open sets in X and let Uθ 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,Τθ) → (Y,Uθ) 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 = {Gi | iI} be an IF open cover of Y. Since f is an IF strongly θ-continuous function, f : (X,Τθ) → (Y,U) is an IF continuous function (Theorem 4.2 of [11]). Therefore, {f?1(Gi) | iI} is an IF θ-open cover of X. Since X is IF θ-compact, there is a finite subset I0 of I such that

    image

    Since f is onto, {Gi | iI0} is a finite subcover of Y. Hence, Y is IF compact.

    (2) Let g = {Gi | iI} be an IF θ-open cover of f(A) in Y. Since f is an IF θ-irresolute, for each Gi, f?1(Gi) is an IF θ-open set. Moreover, {f?1(Gi) | iI} 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(Gi) | iI0}. Therefore, f(A) ≤ {Gi | iI0}. 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 AB is IF θ-compact.

    Proof. Let g = {Gi | iI} be an IF θ-open cover of A B in X. Since Bc is IF θ-open in X, ({Gi | iI}) ∨ Bc 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 ≤ ({Gi | iI0}) ∨ Bc. Therefore, AB ≤ ({Gi | iI0}). Hence, AB is IF θ-compact.

    5. Conclusion

    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.

      >  Conflict of Interest

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

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