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Intuitionistic Fuzzy Topology and Intuitionistic Fuzzy Preorder
Intuitionistic Fuzzy Topology and Intuitionistic Fuzzy Preorder
International Journal of Fuzzy Logic and Intelligent Systems. 2015. Mar, 15(1): 79-86
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 : December 24, 2014
  • Accepted : March 14, 2015
  • Published : March 25, 2015
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Sang Min Yun
Seok Jong Lee

Abstract
This paper is devoted to finding relationship between intuitionistic fuzzy preorders and intuitionistic fuzzy topologies. For any intuitionistic fuzzy preordered space, an intuitionistic fuzzy topology will be constructed. Conversely, for any intuitionistic fuzzy topological space, we obtain an intuitionistic fuzzy preorder on the set. Moreover, we will show that the family of all intuitionistic fuzzy preorders on an underlying set has a very close link to the family of all intuitionistic fuzzy topologies on the set satisfying some extra condition.
Keywords
1. Introduction
The theory of rough sets was introduced by Pawlak [1] . It is an extension of set theory for the research of intelligent systems characterized by insufficient and incomplete information. The relations between rough sets and topological spaces have been studied in some papers [2 4] . It is proved that the pair of upper and lower approximation operators is a pair of closure and interior of a topological space under a crisp reflexive and transitive relation.
The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations in [5] . Furthermore in [6] , the axiomatic approach for fuzzy rough sets were provided. In [7] , the authors presented a general framework for the research of fuzzy rough sets in which both constructive and axiomatic approaches are used.
In [8] , the authors showed that there is one to one correspondence between the family of fuzzy preorders on a nonempty set and the family of fuzzy topologies on this set satisfying certain extra conditions, and hence they are essentially equivalent.
This paper is devoted to finding relationship between intuitionistic fuzzy preorders and intuitionistic fuzzy topologies. For any intuitionistic fuzzy preordered space, an intuitionistic fuzzy topology will be constructed. Conversely, for any intuitionistic fuzzy topological space, we obtain an intuitionistic fuzzy preorder on the set. Moreover, we will show that the family of all intuitionistic fuzzy preorders on an underlying set has a very close link to the family of all intuitionistic fuzzy topologies on the set satisfying some extra condition.
2. Preliminaries
Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair
A = ( µA , νA ).
where the functions µA : X I and νA : X I denote the degree of membership and the degree of nonmembership, respectively and µA + νA ≤ 1 (See [9] ). Obviously every fuzzy set µ in X is an intuitionistic fuzzy set of the form
PPT Slide
Lager Image
.
Throughout this paper, I I denotes the family of all intuitionistic fuzzy numbers ( a , b ) such that a , b ∈ [0, 1] and a + b ≤ 1, with the order relation defined by
( a , b ) ≤ ( c , d ) iff a c and b d .
And IF( X ) denotes the family of all intuitionistic fuzzy sets in X , and ‘IF’ stands for ‘intuitionistic fuzzy.’
Any IF set A = ( µA , νA ) on X can be naturally written as a function A : X I I defined by A ( x ) = ( µA ( x ), νA ( x )) for any x X .
For any IF set A = ( µA , νA ) of X , the value
πA ( x ) = 1 − µA ( x ) − νA ( x )
is called an indeterminancy degree (or hesitancy degree ) of x to A (See [9] ). Szmidt and Kacprzyk [10] call πA ( x ) an intuitionistic index of x in A . Obviously
0 ≤ πA ( x ) ≤ 1, ∀ x X .
Note πA ( x ) = 0 iff νA ( x ) = 1 − µA ( x ). Hence any fuzzy set µA can be regarded as an IF set ( µA , νA ) with πA = 0.
Definition 2.1 ( [11] ). An IF set R on X × X is called an IF relation on X . Moreover, R is called
  • (i)reflexiveifR(x,x) = (1, 0) for allx∈X,
  • (ii)symmetricifR(x,y) =R(y,x) for allx,y∈X,
  • (iii)transitiveifR(x,y) ∧R(y,z) ≤R(x,z) for allx,y,z∈X.
A reflexive and transitive IF relation is called an IF preorder . A symmetric IF preorder is called an IF equivalence . An IF preorder on X is called an IF partial order if for any x , y X , R ( x , y ) = R ( y , x ) = (1, 0) implies that x = y . In this case, ( X , R ) is called an IF partially ordered space . An IF preorder R is called an IF equality if R is both an IF equivalence and an IF partial order.
Remark 2.2. R −1 is called the inverse of R if R −1 ( x , y ) = R ( y , x ) for any x , y X . If R is an IF preorder, so is R −1 . RC is called the complement of R if RC ( x , y ) = ( ν R(x,y) , µ R(x,y) ) where R ( x , y ) = ( µ R(x,y) , ν R(x,y) ). It is obvious that R −1 RC .
Definition 2.3 ( [12] ). Let R be an IF relation on X . Then the two functions
PPT Slide
Lager Image
,
PPT Slide
Lager Image
: IF( X ) → IF( X ), defined by
PPT Slide
Lager Image
are called the upper approximation operator and the lower approximation operator on X , respectively. Moreover, ( X , R ) is called an IF approximation space .
For any IF number ( a , b ) ∈ I I ,
PPT Slide
Lager Image
is an IF set which has the membership value a constant ” a ” and the nonmembership value a constant ” b ” for all x X .
Proposition 2.4 ( [12 , 13] ). Let ( X , R ) be an IF approximation space. Let A , B ∈ IF( X ), { Aj | j J } ⊆ IF( X ) and ( a , b ) ∈ I I . Then we have
PPT Slide
Lager Image
Theorem 2.5 ( [12 , 13] ). Let ( X , R ) be an IF approximation space. Then
(1) R is reflexive
PPT Slide
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(2) R is transitive
PPT Slide
Lager Image
Theorem 2.6. Let ( X , R ) be a reflexive IF approximation space. If
PPT Slide
Lager Image
for each j J , then
PPT Slide
Lager Image
.
Proof By the reflexivity of R and Theorem 2.5,
PPT Slide
Lager Image
. By Proposition 2.4,
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
.
Example 2.7. Let X = { x 1 , x 2 }. Let R = {<( x 1 , x 1 ), 1, 0>, <( x 1 , x 2 ), 0.2, 0.5>,<( x 2 , x 1 ), 0.4, 0.3>,<( x 2 , x 2 ), 1, 0>}. Then ( X , R ) is an IF approximation space. Let A = {< x 1 , 0.6, 0.3>, < x 2 , 0.5, 0.4>} be an IF set on X , then
PPT Slide
Lager Image
Similarly, we obtain
PPT Slide
Lager Image
Hence,
PPT Slide
Lager Image
Similarly, we have
PPT Slide
Lager Image
Proposition 2.8 ( [12] ). For an IF relation R on X and A ∈ IF( X ), the pair
PPT Slide
Lager Image
and
PPT Slide
Lager Image
are “dual”, i.e.,
PPT Slide
Lager Image
where AC is the complement of A .
Definition 2.9 ( [13 , 14] ). An IF topology T on X in the sense of Lowen [15] is a family of IF sets in X that is closed under arbitrary suprema and finite infima, and contains all constant IF sets. The IF sets in T are called open , and their complements, closed .
Definition 2.10 ( [8] ). A Kuratowski IF closure operator on X is a function k : IF( X ) → IF( X ) satisfying for ( a , b ) ∈ I I , A , B ∈ IF( X ),
  • (i),
  • (ii)A≤k(A),
  • (iii)k(A∨B) =k(A) ∨k(B),
  • (iv)k(k(A)) =k(A).
A Kuratowski IF closure operator k on X is called saturated if for all Aj ∈ IF( X ), j J ,
PPT Slide
Lager Image
Furthermore, an IF topology is called saturated if it has a saturated IF closure operator.
Remark 2.11 ( [13] ). Every Kuratowski IF closure operator k on X gives rise to an IF topology on X in which an IF set B is closed iff k ( B ) = B .
3. Intuitionistic Fuzzy Implication Operator
Generally, for ( a 1 , a 2 ),( b 1 , b 2 ) ∈ I I , the implication operator (or residual implicator ) [16 , 17] is defined as follows;
( a 1 , a 2 ) → ( b 1 , b 2 ) = sup{( d 1 , d 2 ) ∈ I I | ( a 1 , a 2 ) ∧ ( d 1 , d 2 ) ≤ ( b 1 , b 2 )}.
If given IF numbers ( a 1 , a 2 ),( b 1 , b 2 ) are comparable, then the IF implication operator is clearly given by
PPT Slide
Lager Image
for all ( a 1 , a 2 ),( b 1 , b 2 ) ∈ I I .
But it is not always able to compare given IF numbers. Nevertheless, in many papers the IF implication operator is studied where given IF numbers are comparable. In this paper, we consider the IF implication operator to the extend that the given IF numbers are not comparable with some restrictions.
Remark 3.1. The IF residual implicator is clearly given by
PPT Slide
Lager Image
for all ( a 1 , a 2 ),( b 1 , b 2 ) ∈ I I .
Definition 3.2. (1) For ( a 1 , a 2 ) ∈ I I , A ∈ IF( X ) and x X , the map (( a 1 , a 2 ) ∧ A ) : I I × IF( X ) → IF( X ) is defined by
(( a 1 , a 2 ) ∧ A )( x ) = ( a 1 , a 2 ) ∧ A ( x ).
(2) For ( a 1 , a 2 ) ∈ I I , A ∈ IF( X ) and x X , the map (( a 1 , a 2 ) → A ) : I I × IF( X ) → IF( X ) is defined by
(( a 1 , a 2 ) → A )( x ) = ( a 1 , a 2 ) → A ( x ).
(3) For A , B ∈ IF( X ), the map ( A B ) : IF( X ) × IF( X ) → IF( X ) is defined by
( A B )( x ) = A ( x ) → B ( x ), ∀ x X .
Remark 3.3. (1) We denote by [( a 1 , a 2 ),(1, 0)] the rectangular plane which represents [ a 1 , 1]×[0, a 2 ]. For a set A X , an IF set χA : X I I is a map defined by
PPT Slide
Lager Image
From the above, if ( a 1 , a 2 ) ∈ I I and A ∈ IF( X ) are comparable, then we have that
( a 1 , a 2 ) → A = χ A−1[(a1, a2),(1,0)] A .
(2) If ( a 1 , a 2 ),( b 1 , b 2 ),( c 1 , c 2 ) ∈ I I are all comparable, then
( a 1 , a 2 ) ∧ ( b 1 , b 2 ) ≤ ( c 1 , c 2 ) iff ( b 1 , b 2 ) ≤ ( a 1 , a 2 ) → ( c 1 , c 2 ).
Clearly the following holds;
(( a 1 , a 2 ) → ( b 1 , b 2 )) ∧ (( b 1 , b 2 ) → ( c 1 , c 2 )) ≤ (( a 1 , a 2 ) → ( c 1 , c 2 )).
Theorem 3.4. The implication operator ”→” is an IF preorder on I I .
Proof By the above property and the fact
( a 1 , a 2 ) → ( a 1 , a 2 ) = (1, 0),
it follows.
4. Intuitionistic Fuzzy Preorder and Intuitionistic Fuzzy Topology
Let ( X , ≤) be a preordered space and A X . Let ↑ A = { y X | y x , for some x A }. If ↑ A = A , then A is called an upper set . Dually if B =↓ B = { y X | y x , for some x B }, then B is called a lower set . The family of all the upper sets of X is clearly a topology on X , which is called the Alexandrov topology (See [18] ) on X , and denoted Γ(≤). We write simply Γ( X ) for the topological space ( X , Γ(≤)).
On the other hand, for a topological space ( X , T ) and x , y X , let x y if x U implies y U for any open set U of X , or equivalently,
PPT Slide
Lager Image
. Then ≤ is a preorder on X , called the specialization order (See [18] ) on X . Denote this preorder by Ω( T ). We also write simply Ω( X ) for ( X , Ω( T )).
A function f : ( X , ≤ 1 ) → ( Y , ≤ 2 ) between two preordered sets is called order-preserving if x 1 y implies f ( x ) ≤ 2 f ( y ).
From now on we are going to enlarge the above ideas to the IF theories in a natural way.
Definition 4.1. Let ( X , R ) be an IF preordered space. Then A ∈ IF( X ) is called an IF upper set in ( X , R ) if
A ( x ) ∧ R ( x , y ) ≤ A ( y ), ∀ x , y X .
Dually, A is called an IF lower set if A ( y ) ∧ R ( x , y ) ≤ A ( x ) for all x , y X .
Let R be an IF preorder on X . For x , y X , the real number R ( x , y ) can be interpreted as the degree to which x is less than or equal to y . The condition A ( x ) ∧ R ( x , y ) ≤ A ( y ) can be interpreted as the statement that if x is in A and x y then y is in A . Particularly, if R is an IF equivalence relation, then an IF set A is an upper set in ( X , R ) if and only if it is a lower set in ( X , R ).
The classical preorder relation x y can be naturally extended to R ( x , y ) = (1, 0) in IF preorder relation. Since (1, 0) = R ( x , y ) ≤ A ( x ) → A ( y ), A ( x ) ≤ A ( y ) for any IF upper set A . That is, x y means A ( x ) ≤ A ( y ). Obviously, the notion of IF upper sets and IF lower sets agrees with that of upper sets and lower sets in classical preordered space.
Definition 4.2. A function f : ( X , R 1 ) → ( Y , R 2 ) between IF preordered spaces is called order-preserving if
R 1 ( x , y ) ≤ R 2 ( f ( x ), f ( y )), ∀ x , y X .
Definition 4.3. Let X be a totally ordered set. An IF set A on X is called
  • (i)increasingif for allx,y∈Xwithx
  • (ii)decreasingif for allx,y∈Xwithx
  • (iii)monotoneifAis increasing or decreasing.
Definition 4.4. Let A = ( µA , νA ) be an IF set. A is called simple if µA or νA is a constant function.
Remark 4.5. (1) If indeterminancy degree πA is a constant function t , then for any two different elements x , y X , πA ( x ) = πA ( y ) = t . So, if µA ( x ) ≤ µA ( y ), then νA ( x ) ≥ νA ( y ). Hence A ( x ) ≤ A ( y ). Therefore A ( x ) and A ( y ) are comparable for any x , y X provided that the hesitancy degree of an IF set A is a constant function.
(2) Suppose that the universal set X is a totally ordered set. If A is monotone or simple, then A ( x ) and A ( y ) are comparable for any different element x , y X .
From now on, in order to avoid getting imprecise value in acting with the implication operator, we will consider only the IF set A : X I I such that A ( x ) and A ( y ) are comparable for any x , y X . By Remark 4.5, any function which is monotone or simple or of constant hesitancy degree is an example of such function.
Lemma 4.6. For a given IF preordered space ( X , R ), an IF set B : X I I is an IF upper set of ( X , R ) if and only if B : ( X , R ) → ( I I , →) is an order-preserving function.
Proof For any B ∈ IF( X ), we have the following relations.
B is an IF upper set in ( X , R )
  • ⇔B(x) ∧R(x,y) ≤B(y) for allx,y∈X
  • ⇔R(x,y) ≤B(x) →B(y) for allx,y∈X
  • ⇔ The mapB: (X,R) → (I⊗I, →) defined bypreserves order.
Theorem 4.7. If ( X , R ) is an IF preordered space, then the family T of all the upper sets in X satisfies the following conditions, and hence it is an IF topology on X .
For any IF sets Aj , A T ;
  • (i),
  • (ii),
  • (iii),
  • (iv),
  • (v) Suppose that for anyA∈Tandx,y∈Xand (a,b) ∈I⊗I,A(x) andA(y) and (a,b) are all comparable. Then ((a,b) →A) ∈T.
Proof (i) Let ( a , b ) ∈ I I , then
PPT Slide
Lager Image
. So
PPT Slide
Lager Image
.
(ii)
PPT Slide
Lager Image
.
(iii)
PPT Slide
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.
(iv) If A is an upper set, then A ( x ) ∧ R ( x , y ) ≤ A ( y ) for any x , y X . So, we obtain that
(( a , b ) ∧ A ( x )) ∧ R ( x , y ) ≤ (( a , b ) ∧ A ( y )),
for all ( a , b ) ∈ I I . This means that (( a , b ) ∧ A ) ∈ T .
(v) By Remark 3.3, (( a , b ) → A ( x )) ∧ ( A ( x ) → A ( y )) ≤ (( a , b ) → A ( y )). Thus(( a , b ) → A ( x )) → (( a , b ) → A ( y )) ≥ ( A ( x ) → A ( y )) ≥ R ( x , y ). Hence (( a , b ) → A ( x )) ∧ R ( x , y ) ≤ (( a , b ) → A ( y )).
Definition 4.8. For an IF preordered space ( X , R ), let
PPT Slide
Lager Image
If T is a family of IF sets in X which satisfies conditions (i)-(v) of Theorem 4.7, then there exists an IF preorder R on X such that T consists of all the upper sets of the IF preordered space ( X , R ). It will be shown in the following theorem.
Lemma 4.9. Let Λ be a subfamily of IF( X ) such that for any A ∈ Λ and for any x , y X , A ( x ) and A ( y ) are comparable. Let
PPT Slide
Lager Image
. Then R is an IF preorder.
PPT Slide
Lager Image
Theorem 4.10. Let Λ be a subfamily of IF( X ) satisfying (i)-(v) of Theorem 4.7 such that for any A ∈ Λ and for any x , y X , A ( x ) and A ( y ) are comparable. Then there exists an IF preorder R such that Λ is the family T of all upper sets in X with respect to R .
Proof Suppose that Λ ⊆ IF( X ) satisfy the conditions (i)-(v) of Theorem 4.7. Define
PPT Slide
Lager Image
. By the above lemma, R is an IF preorder on X . Let B ∈ Λ. Since B ( x ) and B ( y ) are comparable for any x , y X , we have R ( x , y ) ≤ B ( x ) → B ( y ). Hence B ( x ) ∧ R ( x , y ) ≤ B ( y ), i.e. B is an IF upper set. Thus Λ ⊆ T .
What remains is to show that T ⊆ Λ. Take D T . For a given x X , define mx : X I I by mx ( z ) = D ( x ) ∧ R ( x , z ) for all z X . Then mx ( x ) = D ( x ) and mx ( z ) ≤ D ( z ) for all z X . Thus
PPT Slide
Lager Image
.
For each B ∈ Λ and previously given x , define gB : X I I by gB ( z ) = B ( x ) → B ( z ) for all z X . By (v), gB ∈ Λ. By (iii), we obtain that
PPT Slide
Lager Image
. Since
PPT Slide
Lager Image
we have mx ∈ Λ. Note that
PPT Slide
Lager Image
. Therefore D ∈ Λ by (ii).
Example 4.11. Let X = [0, 1] be the universal set. Let Λ be the family of all constant IF sets on X and the IF set A = ( µA , νA ), where µA ( x ) = 1− x , νA ( x ) = x . Take Γ by arbitrary suprema and arbitrary infima with members of Λ. Then clearly Γ ⊆ IF( X ) and it satisfies (i)-(v) of Theorem 4.7. We can define the order R on X by
PPT Slide
Lager Image
Then clearly R is reflexive. Take x , y X such that x y , then R ( x , y ) = A ( y ) and R ( y , x ) = (1, 0). So we know that R is transitive. Therefore R is a preorder on X . Consider A ( x )∧ R ( x , y ) when x y . Then R ( x , y ) = A ( y ), and hence A ( x ) ∧ R ( x , y ) = A ( x ) ∧ A ( y ) ≤ A ( y ). Consequently we know that Γ is the family of all upper sets in X with respect to R .
The following result relates lower sets and upper sets in an IF preordered space ( X , R ) with the upper approximation operator
PPT Slide
Lager Image
and
PPT Slide
Lager Image
, respectively.
Proposition 4.12. Let ( X , R ) be an IF preordered space and A ∈ IF( X ). Then A is a lower set iff
PPT Slide
Lager Image
.
Proof A is a lower set
PPT Slide
Lager Image
Proposition 4.13. Let ( X , R ) be an IF preordered space and A ∈ IF( X ). Then A is an upper set iff
PPT Slide
Lager Image
.
Proof A is an upper set
PPT Slide
Lager Image
Remark 4.14. Let T be the family of all upper sets of an IF preordered space ( X , R ). By Theorem 4.7, T is an Alexandrov IF topology on X . Furthermore
PPT Slide
Lager Image
by the above proposition.
Proposition 4.15. Let ( X , R ) be an IF preordered space. Then
PPT Slide
Lager Image
is an Alexandrov IF topology on X .
Proof (1) Take ( a , b ) ∈ I I , then we know that
PPT Slide
Lager Image
. So
PPT Slide
Lager Image
. Clearly
PPT Slide
Lager Image
. Hence
PPT Slide
Lager Image
. Therefore
PPT Slide
Lager Image
for any ( a , b ) ∈ I I .
(2) Take
PPT Slide
Lager Image
. Then
PPT Slide
Lager Image
for each j J . So
PPT Slide
Lager Image
. Thus
PPT Slide
Lager Image
.
(3) Take
PPT Slide
Lager Image
. Then
PPT Slide
Lager Image
for each j J . So
PPT Slide
Lager Image
. Thus
PPT Slide
Lager Image
.
Remark 4.16. The IF topology
PPT Slide
Lager Image
is dual to the IF topology
PPT Slide
Lager Image
. It follows from the fact that for
PPT Slide
Lager Image
. In addition, the IF topologies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
are Alexandrov IF topologies.
Proposition 4.17. Let ( X , R ) be an IF preordered space and A ∈ IF( X ). Then
  • (i)is the IF interior operator for the IF topology,
  • (ii)is the IF closure operator for the IF topology.
Proof (i) We will show that
PPT Slide
Lager Image
for any A ∈ IF( X ). Since
PPT Slide
Lager Image
,
PPT Slide
Lager Image
. Since
PPT Slide
Lager Image
,
PPT Slide
Lager Image
. By Theorem 2.6,
PPT Slide
Lager Image
. So
PPT Slide
Lager Image
. On the other hand, by
PPT Slide
Lager Image
, we obtain
PPT Slide
Lager Image
. Hence
PPT Slide
Lager Image
. Therefore
PPT Slide
Lager Image
.
(ii) We will show that
PPT Slide
Lager Image
for any A ∈ IF( X ). Since
PPT Slide
Lager Image
,
PPT Slide
Lager Image
. By the duality,
PPT Slide
Lager Image
. Hence
PPT Slide
Lager Image
.
Proposition 4.18. Let k be a saturated IF closure operator on X . Then there exists an IF preorder R on X such that k =
PPT Slide
Lager Image
iff
  • (i)for anyAj∈ IF(X), and
  • (ii)k((a,b) ∧A) = (a,b) ∧k(A) for anyA∈ IF(X) and (a,b) ∈I⊗I.
Proof Suppose that k satisfies (i) and (ii). By using k , we define an IF relation R on X as
R ( x , y ) = k ( χ {y} )( x ), x , y X .
For each A ∈ IF( X ), if x y , we have
( χ {y} A ( y ))( x ) = ( χ {y} )( x )∧ A ( y ) = (0, 1)∧ A ( y ) = (0, 1).
So
PPT Slide
Lager Image
, hence
PPT Slide
Lager Image
.
For every x X , we have
PPT Slide
Lager Image
which implies
PPT Slide
Lager Image
.
Conversely suppose the assumptions. Since k is a Kuratowski IF closure operator, T = { AC ∈ IF( X ) | k ( A ) = A } is an IF topology T on X , and it satisfies (i). By Lemma 4.9, there exists an IF preorder R on X with respect to the family T . Since
PPT Slide
Lager Image
, k satisfies (ii) by Proposition 2.4.
Conflict of InterestNo potential conflict of interest relevant to this article was reported.
BIO
Sang Min Yun is pursuing a doctorate at Chungbuk National University. His research interests include general topology and fuzzy topology. He is a member of KIIS and KMS.
E-mail: jivesm@naver.com
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 Univerisity from 2003 to 2004. His research interests include general topology and fuzzy topology. He is a member of KIIS, KMS, and CMS. He served as a general chair of the 12th International Symposium on Advanced Intelligent Systems(ISIS 2011).
E-mail: sjl@cbnu.ac.kr
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