Categorical Aspects of Intuitionistic Fuzzy Topological Spaces

  • cc icon
  • ABSTRACT

    In this paper, we obtain two types of adjoint functors between the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense, and the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense. Also, we reveal that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense.


  • KEYWORD

    intuitionistic fuzzy topology

  • 1. Introduction

    Chang [2] defined fuzzy topological spaces with the concept of fuzzy set introduced by Zadeh [11]. After that, many generalizations of the fuzzy topology were studied by several authors like Ŝostak [10], Ramadan [9], and Chattopadhyay and his colleagues [3].

    On the other hand, the concept of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of fuzzy sets. Çoker [4] introduced intuitionistic fuzzy topological spaces by using intuitionistic fuzzy sets. Mondal and Samanta [7] introduced the concept of intuitionistic gradation of openness as a generalization of a smooth topology of Ramadan (see [9]). Also, using the idea of degree of openness and degree of nonopenness, Çoker and Demirci [5] defined intuitionistic fuzzy topological spaces in Ŝostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces.

    Lee and Lee [6] revealed that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Çoker’s sense. Also, Park and his colleagues [8] showed that the category of intuitionistic fuzzy topological spaces in Çoker’s sense is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense.

    The aim of this paper is to continue this investigation of categorical relationships between those categories. We obtain two types of adjoint functors between the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense, and the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense. Also, we reveal that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense.

    2. Preliminaries

    We will denote the unit interval [0, 1] of the real line by I. A member μ of I X is called a fuzzy set in X. By and we denote the constant fuzzy sets in X with value 0 and 1, respectively. For any μIX, μc denotes the complement .

    Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair

    image

    where the functions μA : XI and γA : XI denote the degree of membership and the degree of nonmembership, respectively and μA + γA ≤ 1. By and we denote the constant intuitionistic fuzzy sets with value (0, 1) and (1, 0), respectively. Obviously every fuzzy set μ in X is an intuitionistic fuzzy set of the form (μ, ).

    Let f be a mapping from a set X to a set Y . Let A = (µA, γA) be an intuitionistic fuzzy set in X and B = (µB, γB) an intuitionistic fuzzy set in Y . Then

    (1) The image of A under f, denoted by f(A), is an intuitionistic fuzzy set in Y defined by

    image

    (2) The inverse image of B under f, denoted by f−1 (B), is an intuitionistic fuzzy set in X defined by

    image

    All other notations are standard notations of fuzzy set theory.

    Definition 2.1. ( [2]) A Chang’s fuzzy topology on X is a family T of fuzzy sets in X which satisfies the following properties:

    image

    The pair (X, T) is called a fuzzy topological space.

    Definition 2.2. ( [9]) A smooth topology on X is a mapping T : IXI which satisfies the following properties:

    image

    The pair (X, T) is called a smooth topological space.

    Definition 2.3. ( [4]) An intuitionistic fuzzy topology on X is a family T of intuitionistic fuzzy sets in X which satisfies the following properties:

    image

    The pair (X, T) is called an intuitionistic fuzzy topological space.

    Let I(X) be a family of all intuitionistic fuzzy sets in X and let II be the set of the pair (r, s) such that r, sI and r + s ≤ 1.

    Definition 2.4. ( [5]) Let X be a nonempty set. An intuitionistic fuzzy topology in Ŝostak’s sense (SoIFT for short) 𝒯 = (𝒯1, 𝒯2) on X is a mapping 𝒯 : I(X) → II which satisfies the following properties:

    image

    Then (X, 𝒯) = (X, 𝒯1, 𝒯2) is said to be an intuitionistic fuzzy topological space in Ŝostak’s sense (SoIFTS for short). Also, we call 𝒯1(A) the gradation of openness of A and 𝒯2(A) the gradation of nonopenness of A.

    Definition 2.5. ([5]) Let f : (X, 𝒯1, 𝒯2) → (Y, 𝒰1, 𝒰2) be a mapping from a SoIFTS X to a SoIFTS Y. Then f is said to be SoIF continuous if 𝒯1(f−1(B)) ≥ 𝒯1(B) and 𝒯2(f−1(B)) ≤ 𝒯2(B) for each B I(Y ).

    Let (X, 𝒯) be a SoIFTS. Then for each (r, s) ∈ II, the family 𝒯(r,s) defined by

    image

    is an intuitionistic fuzzy topology on X. In this case, 𝒯(r,s) is called the (r, s)-level intuitionistic fuzzy topology on X.

    Let (X, T) be an intuitionistic fuzzy topological space. Then for each (r, s) ∈ II, a SoIFT T(r,s) : I(X) → II defined by

    image

    In this case, T (r,s) is called an (r, s)-th graded SoIFT on X and (X, T(r,s)) is called an (r, s)-th graded SoIFTS on X.

    Definition 2.6. ([7]) Let X be a nonempty set. An intuitionistic fuzzy topology in Mondal and Samanta’s sense(MSIFT for short) T = (T1, T2) on X is a mapping T : IXI I which satisfy the following properties:

    image

    Then (X, T) is said to be an intuitionistic fuzzy topological space in Mondal and Samanta’s sense(MSIFTS for short). T1 and T2 may be interpreted as gradation of openness and gradation of nonopenness, respectively.

    Definition 2.7. ([7]) Let f : (X, T1, T2) → (Y, U1, U2) be a mapping. Then f is said to be MSIF contiunous if T1(f−1(η)) ≥ U1(η) and T2(f−1(η)) ≤ U2(η) for each ηIY.

    Let (X, T) be a MSIFTS. Then for each (r, s) ∈ II, the family T(r,s) defined by

    image

    is a Chang’s fuzzy topology on X. In this case, T(r,s) is called the (r, s)-level Chang’s fuzzy topology on X.

    Let (X, T) be a Chang’s fuzzy topological spaces. Then for each (r, s) ∈ II, a MSIFT T(r,s) : IXI I is defined by

    image

    In this case, T(r,s) is called an (r, s)-th graded MSIFT on X and (X, T(r,s)) is called an (r, s)-th graded MSIFTS on X.

    3. The categorical relationships between MSIFTop and SoIFTop

    Let MSIFTop be the category of all intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense and MSIF continuous mappings, and let SoIFTop be the category of all intuitionistic fuzzy topological spaces in Ŝostak’s sense and SoIF continuous mappings.

    Theorem 3.1. Define a functor F : SoIFTopMSIFTop by F(X, 𝒯) = (X, F(𝒯)) and F(f) = f, where F(𝒯)(η) = (F(𝒯)1(η), F(𝒯)2(η)), F(𝒯)1(η) = ∨ {𝒯1(A) | µA = η}, F(𝒯)2(η) = ⋀ {𝒯2(A) | µA = η}. Then F is a functor.

    Proof. First, we show that F(𝒯) is a MSIFT.

    Clearly, F(𝒯)(η) = F(𝒯)1(η) + F(𝒯)2(η) ≤ 1 for any ηIX.

    image

    (2) Suppose that F(𝒯)1(η ∧ λ) < F(𝒯)1(η) ∧ F(𝒯)1(λ). Then there is a t I such that F(𝒯)1(η∧λ) < t < F(𝒯)1(η) ∧ F(𝒯)1(λ). Since t < F(𝒯)1(η) = ∨ {T1(C) | µC = η}, there is an AI(X) such that t < 𝒯1(A) and µA = η. There is a BI(X) such that t < 𝒯1(B) and µB = λ, because t < F(𝒯)1(λ) = ∨{𝒯1(C) | µC = λ}. Thus t < 𝒯1(A) ∧ 𝒯1(B) and µAB = µAµB = η ∧ λ. Since 𝒯 is a SoIFT, we obtain

    image

    Hence

    image

    This is a contradiction. Thus F(𝒯)1(η ∧ λ) ≥ F(𝒯)1(η) ∧ F(𝒯)2(λ).

    Next, assume that F(𝒯)2(η ∧ λ) > F(𝒯)2(η) ∨ F(𝒯)2(λ). Then there is an sI such that

    image

    Since s > F(𝒯)2(η) = ⋀{𝒯2(C) | µC = η}, there is an AI(X) such that s > 𝒯2(A) and µA = η. As s > F(𝒯)2(λ) = ⋀{𝒯2(C) | µC = λ}, there is a BI(X) such that s > 𝒯2(B) and µB = λ. So s > 𝒯2(A) ∨ 𝒯2(B) and µAB = µAµB = η ∧ λ. Since 𝒯 is a SoIFT, we have s > 𝒯2(A) ∨ 𝒯2(B) ≥ 𝒯2(AB). Thus

    image

    This is a contradiction. Hence F(𝒯)2(η ∧ λ) ≤ F(𝒯)2(η) ∨ F(𝒯)2(λ).

    (3) Suppose that F(𝒯)1(∨ηi) < ⋀F(𝒯)1(ηi). Then there is a tI such that F(𝒯)1(∨ηi) < t < ⋀ F(𝒯)1(ηi). Since t < F(𝒯)1(ηi) = ∨ {T1(C) | µC = ηi} for each i, there is an AiI(X) such that t < 𝒯1(Ai) and µAi = ηi. Thus t ≤ ⋀ 𝒯1(Ai) and µAi = ∨µAi = ∨ηi. As 𝒯 is a SoIFT, we obtain 𝒯1(∪ Ai) ≥ ⋀ 𝒯1(Ai). Hence

    image

    This is a contradiction. Thus F(𝒯)1(∨ηi) ≥ ⋀F(𝒯)1(ηi).

    Next, assume that F(𝒯)2(∨ηi) > ∨F(𝒯)2(ηi). Then there is an sI such that

    image

    Since s > F(𝒯)2(ηi) = ⋀{𝒯2(C) | µC = ηi} for each i, there is a BiI(X) such that s > 𝒯2(Bi) and µBi = ηi. Hence s ≥ ∨𝒯2(Bi) and µBi = ∨µBi = ∨ηi. Since 𝒯 is a SoIFT, we have 𝒯2(∪Bi) ≤ ∨𝒯2(Bi). Thus

    image

    This is a contradiction. Hence F(𝒯)2( ∨ ηi) ≤ ∨F(𝒯)2(ηi). Therefore (X, F(𝒯)) is a MSIFTS.

    Finally, we show that if f : (X, 𝒯) → (Y, 𝒰) is SoIF continuous, then f : (X, F(𝒯)) → (Y, F(𝒰)) is MSIF continuous. Let F(𝒯) = (F(𝒯)1, F(𝒯)2), F(𝒰) = (F(𝒰)1, F(𝒰)2), and λ ∈ IY . Then

    image

    and

    image

    Therefore F is a functor.

    Theorem 3.2. Define a functor G : MSIFTopSoIFTop by G(X, T) = (X, G(T)) and G(f) = f, where G(T)(A) = (G(T)1(A), G(T)2(A)), G(T)1(A) = T1(µA), and G(T)2(A) = T2(µA). Then G is a functor.

    Proof. First, we show that G(T) is a SoIFT.

    Clearly, G(T)1(A) + G(T)2(A) = T1(µA) + T2(µA) ≤ 1 for any AI(X).

    image

    and

    image

    (3) Let AiI(X) for each i. Then

    image

    and

    image

    Hence (X, G(T)) is a SoIFT.

    Next, we show that if f : (X, T) → (Y, U) is MSIF continuous, then f : (X, G(T)) → (Y, G(U)) is SoIF continuous. Let B = (µB, γB) ∈ I(Y ). Then

    image

    and

    image

    Thus f : (X, G(T)) → (Y, G(U)) is SoIF continuous. Consequently, G is a functor.

    Theorem 3.3. The functor G : MSIFTopSoIFTop is a left adjoint of F : SoIFTopMSIFTop.

    Proof. Let (X, T) be an object in MSIFTop and η I X. Then

    image

    Hence lX : (X, T) → FG(X, T) = (X, T) is MSIF continuous.

    Consider (Y, 𝒰) ∈ SoIFTop and a MSIF continuous mapping f : (X, T) → F(Y, 𝒰). In order to show that f : G(X, T) → (Y, 𝒰) is a SoIF continuous mapping, let BI(Y). Then

    image

    and

    image

    Hence f : (X, G(T)1, G(T)2) → (Y, 𝒰1, 𝒰2) is a SoIF continuous mapping. Therefore lX is a G-universal mapping for (X, T) in MSIFTop.

    Theorem 3.4. Define a functor H : SoIFTopMSIFTop by H(X, 𝒯) = (X, H(𝒯)) and H(f) = f, where , and . Then H is a functor.

    Proof. First, we show that H(𝒯) is a MSIFT. Obviously, H(𝒯)(η) = H(𝒯)1(η) + H(𝒯)2(η) ≤ 1 for any ηIX.

    image

    (2) Assume that H(𝒯)1(η ∧ λ) < H(𝒯)1(η) ∧ H(𝒯)1(λ). Then there is a tI such that

    image

    As , there is an AI(X) such that . Since , there is a BI(X) such that and

    image

    Since 𝒯 is a SoIFT, t < 𝒯1(A) ∧ 𝒯1(B) ≤ 𝒯1(AB). Thus

    image

    This is a contradiction. Hence H(𝒯)1(η ∧ λ) ≥ H(𝒯)1(η) ∧ H(𝒯)1(λ).

    Suppose that H(𝒯)2(η∧λ) > H(𝒯)2(η)∨H(𝒯)2(λ). Then there is an sI such that

    image

    Since , there is an A I(X) such that s > 𝒯2(A) and . As , there is a BI(X) such that s > 𝒯2(B) and . So s > 𝒯2(A) ∨ 𝒯2(B) and

    image

    Since 𝒯 is a SoIFT, we obtain s > 𝒯2(A)∨𝒯2(B) ≥ 𝒯2(AB). Hence

    image

    This is a contradiction. Thus H(𝒯)2(η ∧ λ) ≤ H(𝒯)2(η) ∨ H(𝒯)2(λ).

    (3) Assume that H(𝒯)1(∨ηi) < ⋀H(𝒯)1(ηi). Then there is a tI such that

    H(𝒯)1(∨ηi) < t < ⋀H(𝒯)1(ηi).

    As for each i, there is an AiI(X) such that t < 𝒯1(Ai) and . Hence t ≤ ⋀𝒯1(Ai) and

    image

    Since 𝒯 is a SoIFT, we have 𝒯1(∪Ai) ≥ ⋀𝒯1(Ai). Thus

    image

    This is a contradiction. Hence H(𝒯)1(∨ηi) ≥ ⋀H(𝒯)1(ηi).

    Suppose that H(𝒯)2(∨ηi) > ∨H(𝒯)2(ηi). Then there is an sI such that H(𝒯)2(∨ηi) > s > ∨H(𝒯)2(ηi). Since for each i, there is a BiI(X) such that s > 𝒯2(Bi) and . Hence s ≥ ∨𝒯2(Bi) and

    image

    We have 𝒯2(∪Bi) ≤ ∨𝒯2(Bi) because 𝒯 is a SoIFT. Thus

    image

    This is a contradiction. Hence H(𝒯)2( ∨ ηi) ≤ ∨H(𝒯)2(ηi). Therefore (X, H(𝒯)) is a MSIFTS.

    Next, we show that if f : (X, 𝒯) → (Y, 𝒰) is SoIF continuous, then f : (X, H(𝒯)) → (Y, H(𝒰)) is MSIF continuous. Let H(𝒯) = (H(𝒯)1, H(𝒯)2), H(𝒰) = (H(𝒰)1, H(𝒰)2), and ηIX. Then

    image

    and

    image

    Therefore H is a functor.

    Theorem 3.5. Define a functor K : MSIFTopSoIFTop by K(X, T) = (X, K(T)) and K(f) = f, where K(T) = (K(T)1, K(T)2), , and . Then K is a functor.

    Proof. First, we show that K(T) is a SoIFT. Clearly,

    image

    for any AI(X).

    image

    (2) Let A, BI(X). Then

    image

    and

    image

    (3) Let Ai I(X) for each i. Then

    image

    and

    image

    Thus (X, K(T)) is a SoIFTS.

    Finally, we show that if f : (X, T) → (Y, U) is MSIF continuous, then f : (X, K(T)) → (Y, K(U)) is SoIF continuous. Let B = (µB, γB) ∈ I(Y). Then

    image

    and

    image

    Hence f : (X, K(T)) → (Y, K(U)) is SoIF continuous. Consequently, K is a functor.

    Theorem 3.6. The functor K : MSIFTopSoIFTop is a left adjoint of H : SoIFTopMSIFTop.

    Proof. For any (X, T) in MSIFTop and ηIX,

    image

    Hence lX : (X, T) → HK(X, T) = (X, T) is a MSIF continuous mapping. Consider (Y, 𝒰) ∈ SoIFTop and a MSIF continuous mapping f : (X, T) → H(Y, 𝒰). In order to show that f : K(X, T) → (Y, 𝒰) is a SoIF continuous mapping, let BI(Y). Then

    image

    and

    image

    Thus f : (X, K(T)) → (Y, 𝒰) is SoIF continuous. Hence lX is a K-universal mapping for (X, T) in MSIFTop.

    Let (r, s)-gMSIFTop be the category of all (r, s)-th graded intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense and MSIF continuous mappings, and let CFTop be the category of all Chang’s fuzzy topological spaces and fuzzy continuous mappings.

    Theorem 3.7. Two categories CFTop and (r, s)-gMSIFTop are isomorphic.

    Proof. Define F : CFTop → (r, s)-gMSIFTop by F(X, T) = (X, F(T)) and F(f) = f, where

    image

    Define G : (r, s)-gMSIFTopCFTop by G(X, 𝒯) = (X, G(𝒯)) and G(f) = f, where

    G(𝒯) = 𝒯(r,s) = {η ∈ IX | 𝒯1(η) ≥ r and 𝒯2(η) ≤ s}.

    Then F and G are functors. Obviously, GF(T) = G(T(r,s)) = (T(r,s))(r,s) = T and FG(T) = F(𝒯(r,s)) = (𝒯(r,s))(r,s) = 𝒯. Hence CFTop and (r, s)-gMSIFTop are isomorphic.

    Theorem 3.8. The category (r, s)-gMSIFTop is a bireflective full subcategory of MSIFTop.

    Proof. Obviously, (r, s)-gMSIFTop is a full subcategory of MSIFTop. Let (X, T) be an object of MSIFTop. Then for each (r, s) ∈ II, (X, (T(r,s))(r,s)) is an object of (r, s)-gMSIFTop and lX : (X, T) → (X, (T(r,s))(r,s)) is a MSIF continuous mapping. Let (Y, U) be an object of the category (r, s)-gMSIFTop and f : (X, T) → (Y, U) a MSIF continuous mapping. we need only to check that f : (X, (T(r,s))(r,s)) → (Y, U) is a MSIF continuous mapping. Since (Y, U) ∈ (r, s)-gMSIFTop, U(η) = (1, 0), (r, s), or (0, 1). Let U(η) = (1, 0). Then or . In fact,

    image

    and

    image

    In case U(η) = (0, 1), clearly U(η) ≤ (T(r,s))(r,s) (f−1 (η)). Let U(η) = (r, s). Since f : (X, T) → (Y, U) is MSIF continuous, T(f−1 (η)) ≥ U(η) = (r, s). Thus f−1 (η) ∈ T(r,s), and hence (T(r,s))(r,s) (f−1 (η)) = (r, s) = U(η). Therefore f : (X, (T(r,s))(r,s)) → (Y, U) is a MSIF continuous mapping.

    From the above theorems, we have the follwing main result.

    Theorem 3.9. The category CFTop is a bireflective full subcategory of MSIFTop.

    4. Conclusion

    We obtained two types of adjoint functors between the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense, and the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense. Also, we revealed that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense.

    In further research, we will investigate other properties of the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense.

  • 1. Atanassov K. T. 1986 Intuitionistic fuzzy sets [Fuzzy Sets and Systems] Vol.20 P.87-96 google doi
  • 2. Chang C. L. 1968 Fuzzy topological spaces [J. Math. Anal. Appl.] Vol.24 P.182-190 google doi
  • 3. Chattopadhyay K. C., Hazra R. N., Samanta S. K. 1992 Gradation of openness : Fuzzy topology [Fuzzy Sets and Systems] Vol.49 P.237-242 google doi
  • 4. Coker D. 1997 An introduction to intuitionistic fuzzy topological spaces [Fuzzy Sets and Systems] Vol.88 P.81-89 google doi
  • 5. Coker D., Demirci M. 1996 An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense [BUSEFAL] Vol.67 P.67-76 google
  • 6. Lee Seok Jong, Lee Eun Pyo 2000 The category of intuitionistic fuzzy topological spaces [Bull. Korean Math. Soc.] Vol.37 P.63-76 google
  • 7. Mondal Tapas Kumar, Samanta S.K. 2002 On intuitionistic gradation of openness [Fuzzy Sets and Systems] Vol.131 P.323-336 google doi
  • 8. Park Sung Wook, Lee Eun Pyo, Han Hyuk 2003 The category of intutionistic fuzzy topological spaces in ?ostak’s sense [Journal of applied mathematics and computing] Vol.13 P.487-500 google
  • 9. Ramadan A. A. 1992 Smooth topological spaces [Fuzzy Sets and Systems] Vol.48 P.371-375 google doi
  • 10. ?ostak A. P. 1985 On a fuzzy topological structure [Suppl. Rend. Circ. Matem. Janos Palermo, Sr. II] Vol.11 P.89-103 google
  • 11. Zadeh L. A. 1965 Fuzzy sets [Information and Control] Vol.8 P.338-353 google doi
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • [] 
  • []