Categorical Aspects of Intuitionistic Fuzzy Topological Spaces
 Author: Kim Jin Tae, Lee Seok Jong
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 15, Issue2, p137~144, 25 June 2015

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 memberμ ofI ^{X} is called afuzzy set inX . By and we denote the constant fuzzy sets inX with value 0 and 1, respectively. For anyμ ∈I ^{X},μ ^{c} denotes the complement .Let
X be a nonempty set. Anintuitionistic fuzzy set A is an ordered pairwhere the functions
μ_{A} :X →I andγ_{A} :X →I denote the degree of membership and the degree of nonmembership, respectively andμ_{A} +γ_{A} ≤ 1. By and we denote the constant intuitionistic fuzzy sets with value (0, 1) and (1, 0), respectively. Obviously every fuzzy setμ inX is an intuitionistic fuzzy set of the form (μ , ).Let
f be a mapping from a setX to a setY . LetA = (µ_{A}, γ_{A}) be an intuitionistic fuzzy set inX andB = (µ _{B},γ _{B}) an intuitionistic fuzzy set inY . Then(1) The image of
A underf , denoted byf (A ), is an intuitionistic fuzzy set inY defined by(2) The inverse image of
B underf , denoted byf ^{−1} (B ), is an intuitionistic fuzzy set inX defined byAll other notations are standard notations of fuzzy set theory.
Definition 2.1. ( [2]) AChang’s fuzzy topology onX is a familyT of fuzzy sets inX which satisfies the following properties:The pair (
X ,T ) is called afuzzy topological space .Definition 2.2. ( [9]) Asmooth topology onX is a mappingT :I ^{X} →I which satisfies the following properties:The pair (
X ,T ) is called asmooth topological space .Definition 2.3. ( [4]) Anintuitionistic fuzzy topology onX is a familyT of intuitionistic fuzzy sets inX which satisfies the following properties:The pair (
X ,T ) is called anintuitionistic fuzzy topological space. Let
I (X ) be a family of all intuitionistic fuzzy sets inX and letI ⊗I be the set of the pair (r ,s ) such thatr ,s ∈I andr +s ≤ 1.Definition 2.4. ( [5]) LetX be a nonempty set. Anintuitionistic fuzzy topology in Ŝostak’s sense (SoIFT for short) 𝒯 = (𝒯_{1}, 𝒯_{2}) onX 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 anintuitionistic fuzzy topological space in Ŝostak’s sense (SoIFTS for short). Also, we call 𝒯_{1}(A ) thegradation of openness ofA and 𝒯_{2}(A ) thegradation of nonopenness ofA .Definition 2.5. ([5]) Letf : (X , 𝒯_{1}, 𝒯_{2}) → (Y , 𝒰_{1}, 𝒰_{2}) be a mapping from a SoIFTSX to a SoIFTSY . Thenf is said to beSoIF continuous if 𝒯_{1}(f ^{−1}(B )) ≥ 𝒯_{1}(B ) and 𝒯_{2}(f ^{−1}(B )) ≤ 𝒯_{2}(B ) for eachB ∈I (Y ).Let (
X , 𝒯) be a SoIFTS. Then for each (r ,s ) ∈I ⊗I , the family 𝒯_{(r,s)} defined byis an intuitionistic fuzzy topology on
X . In this case, 𝒯_{(r,s)} is called the (r ,s )level intuitionistic fuzzy topology onX .Let (
X ,T ) be an intuitionistic fuzzy topological space. Then for each (r ,s ) ∈I ⊗I , a SoIFTT ^{(r,s)} :I (X ) →I ⊗I defined byIn this case,
T ^{(r,s)} is called an (r ,s )th graded SoIFT onX and (X ,T ^{(r,s)}) is called an (r ,s )th graded SoIFTS onX .Definition 2.6. ([7]) LetX be a nonempty set. Anintuitionistic fuzzy topology in Mondal and Samanta’s sense (MSIFT for short)T = (T _{1},T _{2}) onX is a mappingT :I^{X} →I ⊗I which satisfy the following properties:Then (
X ,T ) is said to be anintuitionistic fuzzy topological space in Mondal and Samanta’s sense (MSIFTS for short).T _{1} andT _{2} may be interpreted asgradation of openness andgradation of nonopenness , respectively.Definition 2.7. ([7]) Letf : (X ,T _{1},T _{2}) → (Y ,U _{1},U _{2}) be a mapping. Thenf is said to beMSIF contiunous ifT _{1}(f ^{−1}(η )) ≥U _{1}(η ) andT _{2}(f ^{−1}(η )) ≤U _{2}(η ) for eachη ∈I ^{Y}.Let (
X ,T ) be a MSIFTS. Then for each (r ,s ) ∈I ⊗I , the familyT _{(r,s)} defined byis a Chang’s fuzzy topology on
X . In this case,T _{(r,s)} is called the (r ,s )level Chang’s fuzzy topology onX .Let (
X ,T ) be a Chang’s fuzzy topological spaces. Then for each (r ,s ) ∈I ⊗I , a MSIFTT ^{(r,s)} :I ^{X} →I ⊗I is defined byIn this case,
T ^{(r,s)} is called an (r ,s )th graded MSIFT onX and (X ,T ^{(r,s)}) is called an (r ,s )th graded MSIFTS onX .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 letSoIFTop be the category of all intuitionistic fuzzy topological spaces in Ŝostak’s sense and SoIF continuous mappings.Theorem 3.1. Define a functorF :SoIFTop →MSIFTop byF (X , 𝒯) = (X ,F (𝒯)) andF (f ) =f , whereF (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 }. ThenF is a functor.Proof . First, we show thatF (𝒯) is a MSIFT.Clearly,
F (𝒯)(η ) =F (𝒯)_{1}(η ) +F (𝒯)_{2}(η ) ≤ 1 for anyη ∈I ^{X}.(2) Suppose that
F (𝒯)_{1}(η ∧ λ) <F (𝒯)_{1}(η ) ∧F (𝒯)_{1}(λ). Then there is at ∈I such thatF (𝒯)_{1}(η ∧λ) <t <F (𝒯)_{1}(η ) ∧F (𝒯)_{1}(λ ). Sincet <F (𝒯)_{1}(η ) = ∨ {T _{1}(C ) µ _{C} =η }, there is anA ∈I (X ) such thatt < 𝒯_{1}(A ) andµ _{A} =η . There is aB ∈I (X ) such thatt < 𝒯_{1}(B ) andµ _{B} = λ, becauset <F (𝒯)_{1}(λ) = ∨{𝒯_{1}(C ) µ _{C} = λ}. Thust < 𝒯_{1}(A ) ∧ 𝒯_{1}(B ) andµ _{A∩B} =µ _{A} ∧µ _{B} =η ∧ λ. Since 𝒯 is a SoIFT, we obtainHence
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 ans ∈I such thatSince
s >F (𝒯)_{2}(η ) = ⋀{𝒯_{2}(C ) µ _{C} =η }, there is anA ∈I (X ) such thats > 𝒯_{2}(A ) andµ _{A} =η . Ass >F (𝒯)_{2}(λ) = ⋀{𝒯_{2}(C ) µ _{C} = λ}, there is aB ∈I (X ) such thats > 𝒯_{2}(B ) andµ _{B} = λ. Sos > 𝒯_{2}(A ) ∨ 𝒯_{2}(B ) andµ _{A∩B} =µ _{A} ∧µ _{B} =η ∧ λ. Since 𝒯 is a SoIFT, we haves > 𝒯_{2}(A ) ∨ 𝒯_{2}(B ) ≥ 𝒯_{2}(A ∩B ). ThusThis 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 at ∈I such thatF (𝒯)_{1}(∨η_{i} ) <t < ⋀F (𝒯)_{1}(η_{i} ). Sincet <F (𝒯)_{1}(η_{i} ) = ∨ {T _{1}(C ) µ_{C} =η_{i} } for eachi , there is anA _{i} ∈I (X ) such thatt < 𝒯_{1}(A _{i}) andµ_{Ai} =η_{i} . Thust ≤ ⋀ 𝒯_{1}(A _{i}) andµ ∪A_{i} = ∨µ_{Ai} = ∨η_{i} . As 𝒯 is a SoIFT, we obtain 𝒯_{1}(∪A _{i}) ≥ ⋀ 𝒯_{1}(A _{i}). HenceThis 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 ans ∈I such thatSince
s >F (𝒯)_{2}(η_{i} ) = ⋀{𝒯_{2}(C ) µ_{C} =η_{i} } for eachi , there is aB _{i} ∈I (X ) such thats > 𝒯_{2}(B _{i}) andµ_{Bi} =η_{i} . Hences ≥ ∨𝒯_{2}(B _{i}) andµ ∪B _{i} = ∨µ_{Bi} = ∨η_{i} . Since 𝒯 is a SoIFT, we have 𝒯_{2}(∪B _{i}) ≤ ∨𝒯_{2}(B _{i}). ThusThis 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, thenf : (X ,F (𝒯)) → (Y ,F (𝒰)) is MSIF continuous. LetF (𝒯) = (F (𝒯)_{1},F (𝒯)_{2}),F (𝒰) = (F (𝒰)_{1},F (𝒰)_{2}), and λ ∈I ^{Y} . Thenand
Therefore
F is a functor.Theorem 3.2. Define a functorG :MSIFTop →SoIFTop byG (X ,T ) = (X ,G (T )) andG (f ) =f , whereG (T )(A ) = (G (T )_{1}(A ),G (T )_{2}(A )),G (T )_{1}(A ) =T _{1}(µ_{A} ), andG (T )_{2}(A ) =T _{2}(µ_{A} ). ThenG is a functor.Proof . First, we show thatG (T ) is a SoIFT.Clearly,
G (T )_{1}(A ) +G (T )_{2}(A ) =T _{1}(µ_{A} ) +T _{2}(µ_{A} ) ≤ 1 for anyA ∈I (X ).and
(3) Let
A _{i} ∈I (X ) for eachi . Thenand
Hence (
X ,G (T )) is a SoIFT.Next, we show that if
f : (X ,T ) → (Y ,U ) is MSIF continuous, thenf : (X ,G (T )) → (Y ,G (U )) is SoIF continuous. LetB = (µ_{B} ,γ_{B} ) ∈I (Y ). Thenand
Thus
f : (X ,G (T )) → (Y ,G (U )) is SoIF continuous. Consequently,G is a functor.Theorem 3.3. The functorG :MSIFTop →SoIFTop is a left adjoint ofF :SoIFTop →MSIFTop .Proof. Let (X ,T ) be an object inMSIFTop andη ∈I ^{X}. ThenHence l_{X} : (
X ,T ) →FG (X ,T ) = (X ,T ) is MSIF continuous.Consider (
Y , 𝒰) ∈SoIFTop and a MSIF continuous mappingf : (X ,T ) →F (Y , 𝒰). In order to show thatf :G (X ,T ) → (Y , 𝒰) is aSoIF continuous mapping, letB ∈I (Y ). Thenand
Hence
f : (X ,G (T )_{1},G (T )_{2}) → (Y , 𝒰_{1}, 𝒰_{2}) is a SoIF continuous mapping. Therefore l_{X} is aG universal mapping for (X ,T ) inMSIFTop .Theorem 3.4. Define a functorH :SoIFTop →MSIFTop byH (X , 𝒯) = (X ,H (𝒯)) andH (f ) =f , where , and . ThenH is a functor.Proof . First, we show thatH (𝒯) is a MSIFT. Obviously,H (𝒯)(η ) =H (𝒯)_{1}(η ) +H (𝒯)_{2}(η ) ≤ 1 for anyη ∈I ^{X}.(2) Assume that
H (𝒯)_{1}(η ∧ λ) <H (𝒯)_{1}(η ) ∧H (𝒯)_{1}(λ). Then there is at ∈I such thatAs , there is an
A ∈I (X ) such that . Since , there is aB ∈I (X ) such that andSince 𝒯 is a SoIFT,
t < 𝒯_{1}(A ) ∧ 𝒯_{1}(B ) ≤ 𝒯_{1}(A ∩B ). ThusThis 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 ans ∈I such thatSince , there is an
A ∈I (X ) such thats > 𝒯_{2}(A ) and . As , there is aB ∈I (X ) such thats > 𝒯_{2}(B ) and . Sos > 𝒯_{2}(A ) ∨ 𝒯_{2}(B ) andSince 𝒯 is a SoIFT, we obtain
s > 𝒯_{2}(A )∨𝒯_{2}(B ) ≥ 𝒯_{2}(A ∩B ). HenceThis 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 at ∈I such thatH(𝒯)1(∨ηi) < t < ⋀H(𝒯)1(ηi).
As for each
i , there is anA _{i} ∈I (X ) such thatt < 𝒯_{1}(A _{i}) and . Hencet ≤ ⋀𝒯_{1}(A _{i}) andSince 𝒯 is a SoIFT, we have 𝒯_{1}(∪
A _{i}) ≥ ⋀𝒯_{1}(A _{i}). ThusThis 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 ans ∈ _{I} such thatH (𝒯)_{2}(∨η_{i} ) >s > ∨H (𝒯)_{2}(η_{i} ). Since for eachi , there is aB _{i} ∈I (X ) such thats > 𝒯_{2}(B _{i}) and . Hences ≥ ∨𝒯_{2}(B _{i}) andWe have 𝒯_{2}(∪
B _{i}) ≤ ∨𝒯_{2}(B _{i}) because 𝒯 is a SoIFT. ThusThis 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, thenf : (X ,H (𝒯)) → (Y ,H (𝒰)) is MSIF continuous. LetH (𝒯) = (H (𝒯)_{1},H (𝒯)_{2}),H (𝒰) = (H (𝒰)_{1},H (𝒰)_{2}), andη ∈I ^{X}. Thenand
Therefore
H is a functor.Theorem 3.5. Define a functorK :MSIFTop →SoIFTop byK (X ,T ) = (X ,K (T )) andK (f ) =f , whereK (T ) = (K (T )_{1},K (T )_{2}), , and . ThenK is a functor.Proof . First, we show thatK (T ) is a SoIFT. Clearly,for any
A ∈I (X ).(2) Let
A ,B ∈I (X ). Thenand
(3) Let
A _{i }∈I (X ) for eachi . Thenand
Thus (
X ,K (T )) is a SoIFTS.Finally, we show that if
f : (X ,T ) → (Y ,U ) is MSIF continuous, thenf : (X ,K (T )) → (Y ,K (U )) is SoIF continuous. LetB = (µ_{B} ,γ_{B} ) ∈I (Y ). Thenand
Hence
f : (X ,K (T )) → (Y ,K (U )) is SoIF continuous. Consequently,K is a functor.Theorem 3.6. The functorK :MSIFTop →SoIFTop is a left adjoint ofH :SoIFTop →MSIFTop .Proof. For any (X ,T ) inMSIFTop andη ∈I ^{X},Hence l_{X} : (
X ,T ) →H K (X ,T ) = (X ,T ) is a MSIF continuous mapping. Consider (Y , 𝒰) ∈SoIFTop and a MSIF continuous mappingf : (X ,T ) →H (Y , 𝒰). In order to show thatf :K (X ,T ) → (Y , 𝒰) is a SoIF continuous mapping, letB ∈I (Y ). Thenand
Thus
f : (X ,K (T )) → (Y , 𝒰) is SoIF continuous. Hence l_{X} is aK universal mapping for (X ,T ) inMSIFTop .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 letCFTop be the category of all Chang’s fuzzy topological spaces and fuzzy continuous mappings.Theorem 3.7. Two categoriesCFTop and (r ,s )gMSIFTop are isomorphic.Proof. DefineF :CFTop → (r ,s )gMSIFTop byF (X ,T ) = (X ,F (T )) andF (f ) =f , whereDefine
G : (r ,s )gMSIFTop →CFTop byG (X , 𝒯) = (X ,G (𝒯)) andG (f ) =f , whereG(𝒯) = 𝒯(r,s) = {η ∈ IX  𝒯1(η) ≥ r and 𝒯2(η) ≤ s}.
Then
F andG are functors. Obviously,GF (T ) =G (T ^{(r,s)}) = (T ^{(r,s)})_{(r,s)} =T andFG (T ) =F (𝒯_{(r,s)}) = (𝒯_{(r,s)})^{(r,s)} = 𝒯. HenceCFTop and (r ,s )gMSIFTop are isomorphic.Theorem 3.8. The category (r ,s )gMSIFTop is a bireflective full subcategory ofMSIFTop. Proof. Obviously, (r ,s )gMSIFTop is a full subcategory ofMSIFTop . Let (X ,T ) be an object ofMSIFTop . 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 andf : (X ,T ) → (Y ,U ) a MSIF continuous mapping. we need only to check thatf : (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). LetU (η ) = (1, 0). Then or . In fact,and
In case
U (η ) = (0, 1), clearlyU (η ) ≤ (T _{(r,s)})^{(r,s)} (f ^{−1} (η )). LetU (η ) = (r ,s ). Sincef : (X ,T ) → (Y ,U ) is MSIF continuous,T (f ^{−1} (η )) ≥U (η ) = (r ,s ). Thusf ^{−1} (η ) ∈T (r ,s ), and hence (T _{(r,s)})^{(r,s)} (f ^{−1} (η )) = (r ,s ) =U (η ). Thereforef : (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 categoryCFTop is a bireflective full subcategory ofMSIFTop .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.

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