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Categorical Aspects of Intuitionistic Fuzzy Topological Spaces
Categorical Aspects of Intuitionistic Fuzzy Topological Spaces
International Journal of Fuzzy Logic and Intelligent Systems. 2015. Jun, 15(2): 137-144
Copyright © 2015, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : May 29, 2015
  • Accepted : June 25, 2015
  • Published : June 25, 2015
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Jin Tae Kim
Seok Jong Lee

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.
Keywords
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
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and
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we denote the constant fuzzy sets in X with value 0 and 1, respectively. For any μ I X , μ c denotes the complement
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.
Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair
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where the functions μA : X I and γA : X I denote the degree of membership and the degree of nonmembership, respectively and μA + γA ≤ 1. By
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and
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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 ,
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).
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
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(2) The inverse image of B under f , denoted by f −1 ( B ), is an intuitionistic fuzzy set in X defined by
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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:
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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:
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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:
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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:
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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
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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
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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 : IX I I which satisfy the following properties:
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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
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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
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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 : 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 ) = ∨ {𝒯 1 ( A ) | µA = η }, F (𝒯) 2 ( η ) = ⋀ {𝒯 2 ( 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 η I X .
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,
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,
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, and
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.
(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 AB = µ A µ B = η ∧ λ. Since 𝒯 is a SoIFT, we obtain
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Hence
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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
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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 AB = 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
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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 t I such that F (𝒯) 1 (∨ ηi ) < t < ⋀ F (𝒯) 1 ( ηi ). Since t < F (𝒯) 1 ( ηi ) = ∨ { T 1 ( C ) | µC = ηi } for each i , there is an A i I ( X ) such that t < 𝒯 1 ( A i ) and µAi = ηi . Thus t ≤ ⋀ 𝒯 1 ( A i ) and µ Ai = ∨ µAi = ∨ ηi . As 𝒯 is a SoIFT, we obtain 𝒯 1 (∪ A i ) ≥ ⋀ 𝒯 1 ( A i ). Hence
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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 s I such that
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Since s > F (𝒯) 2 ( ηi ) = ⋀{𝒯 2 ( C ) | µC = ηi } for each i , there is a B i I ( X ) such that s > 𝒯 2 ( B i ) and µBi = ηi . Hence s ≥ ∨𝒯 2 ( B i ) and µ B i = ∨ µBi = ∨ ηi . Since 𝒯 is a SoIFT, we have 𝒯 2 (∪ B i ) ≤ ∨𝒯 2 ( B i ). Thus
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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 ∈ I Y . Then
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and
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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 ( µA ), and G ( T ) 2 ( A ) = T 2 ( 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 ) = T 1 ( µA ) + T 2 ( µA ) ≤ 1 for any A I ( X ).
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and
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(3) Let A i I ( X ) for each i . Then
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and
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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
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and
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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
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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
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and
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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
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, and
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. 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 .
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,
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,
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, and
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.
(2) Assume that H (𝒯) 1 ( η ∧ λ) < H (𝒯) 1 ( η ) ∧ H (𝒯) 1 (λ). Then there is a t I such that
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As
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, there is an A I ( X ) such that
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. Since
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, there is a B I ( X ) such that
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and
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Since 𝒯 is a SoIFT, t < 𝒯 1 ( A ) ∧ 𝒯 1 ( B ) ≤ 𝒯 1 ( A B ). Thus
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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
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Since
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, there is an A I ( X ) such that s > 𝒯 2 ( A ) and
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. As
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, there is a B I ( X ) such that s > 𝒯 2 ( B ) and
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. So s > 𝒯 2 ( A ) ∨ 𝒯 2 ( B ) and
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Since 𝒯 is a SoIFT, we obtain s > 𝒯 2 ( A )∨𝒯 2 ( B ) ≥ 𝒯 2 ( A B ). Hence
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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 t I such that
  • H(𝒯)1(∨ηi)
As
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for each i , there is an A i I ( X ) such that t < 𝒯 1 ( A i ) and
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. Hence t ≤ ⋀𝒯 1 ( A i ) and
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Since 𝒯 is a SoIFT, we have 𝒯 1 (∪ A i ) ≥ ⋀𝒯 1 ( A i ). Thus
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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 s I such that H (𝒯) 2 (∨ ηi ) > s > ∨ H (𝒯) 2 ( ηi ). Since
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for each i , there is a B i I ( X ) such that s > 𝒯 2 ( B i ) and
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. Hence s ≥ ∨𝒯 2 ( B i ) and
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We have 𝒯 2 (∪ B i ) ≤ ∨𝒯 2 ( B i ) because 𝒯 is a SoIFT. Thus
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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 η I X . Then
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and
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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 ),
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, and
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. Then K is a functor.
Proof . First, we show that K ( T ) is a SoIFT. Clearly,
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for any A I ( X ).
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,
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, and
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.
(2) Let A , B I ( X ). Then
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and
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(3) Let A i I ( X ) for each i . Then
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and
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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
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and
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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 ,
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Hence l X : ( X , T ) → H K ( 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
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and
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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
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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) ≥rand 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
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or
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. In fact,
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and
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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.
Acknowledgements
This work was supported by the research grant of Chungbuk National University in 2013.
BIO
Jin Tae Kim received the Ph. D. degree from Chungbuk National University in 2012. His research interests include general topology and fuzzy topology. He is a member of KIIS and KMS.
E-mail: kjtmath@hanmail.net
Seok Jong Lee received the M. S. and Ph. D. degrees from Yonsei University in 1986 and 1990, respectively. He is a professor at the Department of Mathematics, Chungbuk National University since 1989. He was a visiting scholar in Carleton University from 1995 to 1996, and Wayne State University from 2003 to 2004. His research interests include general topology and fuzzy topology. He is a member of KIIS, KMS, and CMS.
E-mail: sjl@cbnu.ac.kr
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