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 μ ∈ I^{X}, μ^{c} denotes the complement .
Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair
where the functions μ_{A} : X → I and γ_{A} : X → I denote the degree of membership and the degree of nonmembership, respectively and μ_{A} + γ_{A} ≤ 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 = (µ_{A}, γ_{A}) be an intuitionistic fuzzy set in X and B = (µ_{B}, γ_{B}) 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
(2) The inverse image of B under f, denoted by f^{−1} (B), is an intuitionistic fuzzy set in X defined by
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:
The pair (X, T) is called a fuzzy topological space.
Definition 2.2. ( [9]) A smooth topology on X is a mapping T : I^{X} → I which satisfies the following properties:
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:
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 I ⊗ I be the set of the pair (r, s) such that r, s ∈ I 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 = (𝒯_{1}, 𝒯_{2}) on X is a mapping 𝒯 : I(X) → I ⊗ I which satisfies the following properties:
Then (X, 𝒯) = (X, 𝒯_{1}, 𝒯_{2}) is said to be an intuitionistic fuzzy topological space in Ŝostak’s sense (SoIFTS for short). Also, we call 𝒯_{1}(A) the gradation of openness of A and 𝒯_{2}(A) the gradation of nonopenness of A.
Definition 2.5. ([5]) Let f : (X, 𝒯_{1}, 𝒯_{2}) → (Y, 𝒰_{1}, 𝒰_{2}) be a mapping from a SoIFTS X to a SoIFTS Y. Then f is said to be SoIF continuous if 𝒯_{1}(f^{−1}(B)) ≥ 𝒯_{1}(B) and 𝒯_{2}(f^{−1}(B)) ≤ 𝒯_{2}(B) for each B ∈ I(Y ).
Let (X, 𝒯) be a SoIFTS. Then for each (r, s) ∈ I ⊗ I, the family 𝒯_{(r,s)} defined by
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) ∈ I ⊗ I, a SoIFT T^{(r,s)} : I(X) → I ⊗ I defined by
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 = (T_{1}, T_{2}) on X is a mapping T : I^{X} → I ⊗ I which satisfy the following properties:
Then (X, T) is said to be an intuitionistic fuzzy topological space in Mondal and Samanta’s sense(MSIFTS for short). T_{1} and T_{2} may be interpreted as gradation of openness and gradation of nonopenness, respectively.
Definition 2.7. ([7]) Let f : (X, T_{1}, T_{2}) → (Y, U_{1}, U_{2}) be a mapping. Then f is said to be MSIF contiunous if T_{1}(f^{−1}(η)) ≥ U_{1}(η) and T_{2}(f^{−1}(η)) ≤ U_{2}(η) for each η ∈ I^{Y}.
Let (X, T) be a MSIFTS. Then for each (r, s) ∈ I ⊗ I, the family T_{(r,s)} defined by
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) ∈ I ⊗ I, a MSIFT T^{(r,s)} : I^{X} → I ⊗ I is defined by
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.
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 : SoIFTop → MSIFTop 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) | µ_{A} = η}, F(𝒯)_{2}(η) = ⋀ {𝒯_{2}(A) | µ_{A} = 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 η ∈ I^{X}.
(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) = ∨ {T_{1}(C) | µ_{C} = η}, there is an A ∈ I(X) such that t < 𝒯_{1}(A) and µ_{A} = η. There is a B ∈ I(X) such that t < 𝒯_{1}(B) and µ_{B} = λ, because t < F(𝒯)_{1}(λ) = ∨{𝒯_{1}(C) | µ_{C} = λ}. Thus t < 𝒯_{1}(A) ∧ 𝒯_{1}(B) and µ_{A∩B} = µ_{A} ∧ µ_{B} = η ∧ λ. Since 𝒯 is a SoIFT, we obtain
Hence
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 s ∈ I such that
Since s > F(𝒯)_{2}(η) = ⋀{𝒯_{2}(C) | µ_{C} = η}, there is an A ∈ I(X) such that s > 𝒯_{2}(A) and µ_{A} = η. As s > F(𝒯)_{2}(λ) = ⋀{𝒯_{2}(C) | µ_{C} = λ}, there is a B ∈ I(X) such that s > 𝒯_{2}(B) and µ_{B} = λ. So s > 𝒯_{2}(A) ∨ 𝒯_{2}(B) and µ_{A∩B} = µ_{A} ∧ µ_{B} = η ∧ λ. Since 𝒯 is a SoIFT, we have s > 𝒯_{2}(A) ∨ 𝒯_{2}(B) ≥ 𝒯_{2}(A ∩ B). Thus
This is a contradiction. Hence F(𝒯)_{2}(η ∧ λ) ≤ F(𝒯)_{2}(η) ∨ F(𝒯)_{2}(λ).
(3) Suppose that F(𝒯)_{1}(∨η_{i}) < ⋀F(𝒯)_{1}(η_{i}). Then there is a t ∈ I such that F(𝒯)_{1}(∨η_{i}) < t < ⋀ F(𝒯)_{1}(η_{i}). Since t < F(𝒯)_{1}(η_{i}) = ∨ {T_{1}(C) | µ_{C} = η_{i}} for each i, there is an A_{i} ∈ I(X) such that t < 𝒯_{1}(A_{i}) and µ_{Ai} = η_{i}. Thus t ≤ ⋀ 𝒯_{1}(A_{i}) and µ∪ A_{i} = ∨µ_{Ai} = ∨η_{i}. As 𝒯 is a SoIFT, we obtain 𝒯_{1}(∪ A_{i}) ≥ ⋀ 𝒯_{1}(A_{i}). Hence
This is a contradiction. Thus F(𝒯)_{1}(∨η_{i}) ≥ ⋀F(𝒯)_{1}(η_{i}).
Next, assume that F(𝒯)_{2}(∨η_{i}) > ∨F(𝒯)_{2}(η_{i}). Then there is an s ∈ I such that
Since s > F(𝒯)_{2}(η_{i}) = ⋀{𝒯_{2}(C) | µ_{C} = η_{i}} for each i, there is a B_{i} ∈ I(X) such that s > 𝒯_{2}(B_{i}) and µ_{Bi} = η_{i}. Hence s ≥ ∨𝒯_{2}(B_{i}) and µ∪B_{i} = ∨µ_{Bi} = ∨η_{i}. Since 𝒯 is a SoIFT, we have 𝒯_{2}(∪B_{i}) ≤ ∨𝒯_{2}(B_{i}). Thus
This is a contradiction. Hence F(𝒯)_{2}( ∨ η_{i}) ≤ ∨F(𝒯)_{2}(η_{i}). 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 ∈ I^{Y} . Then
and
Therefore F is a functor.
Theorem 3.2. Define a functor G : MSIFTop → SoIFTop 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) = T_{1}(µ_{A}), and G(T)_{2}(A) = T_{2}(µ_{A}). Then G is a functor.
Proof. First, we show that G(T) is a SoIFT.
Clearly, G(T)_{1}(A) + G(T)_{2}(A) = T_{1}(µ_{A}) + T_{2}(µ_{A}) ≤ 1 for any A ∈ I(X).
and
(3) Let A_{i} ∈ I(X) for each i. Then
and
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 = (µ_{B}, γ_{B}) ∈ I(Y ). Then
and
Thus f : (X, G(T)) → (Y, G(U)) is SoIF continuous. Consequently, G is a functor.
Theorem 3.3. The functor G : MSIFTop → SoIFTop is a left adjoint of F : SoIFTop → MSIFTop.
Proof. Let (X, T) be an object in MSIFTop and η ∈ I ^{X}. Then
Hence l_{X} : (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 B ∈ I(Y). Then
and
Hence f : (X, G(T)_{1}, G(T)_{2}) → (Y, 𝒰_{1}, 𝒰_{2}) is a SoIF continuous mapping. Therefore l_{X} is a G-universal mapping for (X, T) in MSIFTop.
Theorem 3.4. Define a functor H : SoIFTop → MSIFTop 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 η ∈ I^{X}.
(2) Assume that H(𝒯)_{1}(η ∧ λ) < H(𝒯)_{1}(η) ∧ H(𝒯)_{1}(λ). Then there is a t ∈ I such that
As , there is an A ∈ I(X) such that . Since , there is a B ∈ I(X) such that and
Since 𝒯 is a SoIFT, t < 𝒯_{1}(A) ∧ 𝒯_{1}(B) ≤ 𝒯_{1}(A ∩ B). Thus
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 s ∈ I such that
Since , there is an A ∈ I(X) such that s > 𝒯_{2}(A) and . As , there is a B ∈ I(X) such that s > 𝒯_{2}(B) and . So s > 𝒯_{2}(A) ∨ 𝒯_{2}(B) and
Since 𝒯 is a SoIFT, we obtain s > 𝒯_{2}(A)∨𝒯_{2}(B) ≥ 𝒯_{2}(A∩B). Hence
This is a contradiction. Thus H(𝒯)_{2}(η ∧ λ) ≤ H(𝒯)_{2}(η) ∨ H(𝒯)_{2}(λ).
(3) Assume that H(𝒯)_{1}(∨η_{i}) < ⋀H(𝒯)_{1}(η_{i}). Then there is a t ∈ I such that
H(𝒯)1(∨ηi) < t < ⋀H(𝒯)1(ηi).
As for each i, there is an A_{i} ∈ I(X) such that t < 𝒯_{1}(A_{i}) and . Hence t ≤ ⋀𝒯_{1}(A_{i}) and
Since 𝒯 is a SoIFT, we have 𝒯_{1}(∪A_{i}) ≥ ⋀𝒯_{1}(A_{i}). Thus
This is a contradiction. Hence H(𝒯)_{1}(∨η_{i}) ≥ ⋀H(𝒯)_{1}(η_{i}).
Suppose that H(𝒯)_{2}(∨η_{i}) > ∨H(𝒯)_{2}(η_{i}). Then there is an s ∈ _{I} such that H(𝒯)_{2}(∨η_{i}) > s > ∨H(𝒯)_{2}(η_{i}). Since for each i, there is a B_{i} ∈ I(X) such that s > 𝒯_{2}(B_{i}) and . Hence s ≥ ∨𝒯_{2}(B_{i}) and
We have 𝒯_{2}(∪B_{i}) ≤ ∨𝒯_{2}(B_{i}) because 𝒯 is a SoIFT. Thus
This is a contradiction. Hence H(𝒯)_{2}( ∨ η_{i}) ≤ ∨H(𝒯)_{2}(η_{i}). 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 η ∈ I^{X}. Then
and
Therefore H is a functor.
Theorem 3.5. Define a functor K : MSIFTop → SoIFTop 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,
for any A ∈ I(X).
(2) Let A, B ∈ I(X). Then
and
(3) Let A_{i }∈ I(X) for each i. Then
and
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 = (µ_{B}, γ_{B}) ∈ I(Y). Then
and
Hence f : (X, K(T)) → (Y, K(U)) is SoIF continuous. Consequently, K is a functor.
Theorem 3.6. The functor K : MSIFTop → SoIFTop is a left adjoint of H : SoIFTop → MSIFTop.
Proof. For any (X, T) in MSIFTop and η ∈ I^{X},
Hence l_{X} : (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 B ∈ I(Y). Then
and
Thus f : (X, K(T)) → (Y, 𝒰) is SoIF continuous. Hence l_{X} 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
Define G : (r, s)-gMSIFTop → CFTop 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) ∈ I ⊗ I, (X, (T(_{r,s}))^{(r,s)}) is an object of (r, s)-gMSIFTop and l_{X} : (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,
and
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.