Int. J. of Mathematical Sciences and Applications, Vol. 1, No. 3, September 2011 Copyright Mind Reader Publications www.journalshub.com
A NEW VIEW ON COMPACTNESS IN SMOOTH FUZZY TOPOLOGICAL SPACES B.Amudhambigai
[email protected] M.K.Uma and E.Roja Department of Mathematics Sri Sarada College for Women, Salem-16 TamilNadu, India. Abstract In this paper, new definitions of fuzzy closure and fuzzy interior of an r-fuzzy set which satisfy almost all properties of the corresponding definitions in smooth fuzzy topological spaces in the sense of Sostak [8] and Ramadan [5] are established. As a consequence of these definitions the concept of g -P-fuzzy compact space is introduced and established. Keywords: spaces.
g -P-fuzzy compact spaces, g -P-fuzzy almost compact spaces and g -P-fuzzy nearly compact
Mathematics Subject Classification (2000): 54A40, 03E72. 1. Introduction and Preliminaries The concept of fuzzy set was introduced by Zadeh [12] in his classical paper. Fuzzy sets have applications in many fields such as information [7] and control [11]. Chang [1] introduced the notion of a fuzzy topology. Later Lowen [3] redefined what is now known as startified fuzzy topology. Sostak [8] introduced the notion of fuzzy topology as an extension of Chang an Lowen’s fuzzy topology. Later on he has developed the theory of fuzzy topological spaces in [9] and [10]. Defining the closure and the interior of a fuzzy set in fuzzy topological space, a different approach for the compactness structure of fuzzy topological space was introduced in [2]. The concept of g -open set was discussed by Rajesh and Erdal
g -P ( T ). In this connection, g -P ( T ) fuzzy -P ( T ) satisfy almost all properties of the closure and g -P ( T ) fuzzy interior of a fuzzy set for which g
Ekici [4]. The purpose of this paper is to study a new family
corresponding definitions in smooth fuzzy topological spaces in the sense of Sostak [8] and Ramadan [5]. As a consequence of these definitions the topological concepts, especially various types of compactness are established. Throughout this paper, let X be a nonempty set, I = [ 0, 1 ] and I0 = ( 0, 1 ]. For I, α ( x ) = for all x X. X Definition 1.1 [8] A function T : I I is called a smooth fuzzy topology on X if it satisfies the following conditions :
= 1,
( 0 ) =T 1
(1)
T
(2)
T ( 1 2 ) T ( 1 ) T ( 2) for any 1, 2 I ,
(3)
T
X
X j T ( j ) for any { j } j I . j j
The pair ( X, T ) is called a smooth fuzzy topological space. X Remark 1.1 Let ( X, T ) be a smooth fuzzy topological space. Then for each r I0, Tr = { I : T ( ) r } is Chang’s fuzzy topology on X.
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B.Amudhambigai, M.K.Uma & E.Roja
X
Definition 1.2 [6] Let ( X, T ) be a smooth fuzzy topological space. For each I , r I0, an operator CT : X X I I0 I is defined as follows: X
CT ( , r ) = { : , T ( 1 ) r }. For , I and r, s I0, it satisfies the following conditions: (1) CT ( 0 , r ) = 0 . (2) CT ( , r ). (3) CT ( , r ) CT ( , r ) = CT ( , r ). (4) CT ( , r ) CT ( , s ), if r s. (5) CT ( CT ( , r ), r ) = CT ( , r ). X Proposition 1.1 [6] Let ( X, T ) be a smooth fuzzy topological space. For each I , r I0, an operator IT : X
X
I I0 I is defined as follows: X IT ( , r ) = { : , T ( ) r }. For , I and r, s I0, it satisfies the following conditions: (1)
IT ( 1 , r ) = 1 CT ( , r ).
(2) IT ( 1 , r ) = 1 . IT ( , r ). (3) (4) IT ( , r ) IT ( , r ) = IT ( , r ). (5) IT ( , r ) IT ( , s ), if r s. (6) IT ( IT ( , r ), r ) = IT ( , r ). Definition 1.3 Let ( X, T ) be a smooth fuzzy topological space. For IX and r I0, is called ˆ -closed if CT ( , r ) whenever µ and is r-fuzzy semi-open. The complement (1) r-fuzzy g
ˆ -closed set is said to be an r-fuzzy gˆ -open set. of an r-fuzzy g ˆ -open. The complement of (2) r-fuzzy *g-closed if CT ( , r ) whenever µ and is r-fuzzy g an r-fuzzy *g-closed set is said to be an r-fuzzy *g-open set. (3) r-fuzzy #g-semi-closed ( briefly, #fgs-closed ) if SCT ( , r ) whenever µ and is r-fuzzy *g-open. The complement of an r-fuzzy #g-semi-closed set is said to be an r-fuzzy #g-semi-open # set ( briefly, r- fgs-open ). (4) r-fuzzy g -closed if CT ( , r ) whenever µ and is r-#fgs-open. The complement of an r -closed set is said to be an r-fuzzy g -open set. fuzzy g -P-FUZZY COMPACTNESS AND CONCERNED RELATIONS 2. VARIOUS TYPES OF g
-P-fuzzy compactness are discussed. Also, some interesting In this section, various types of g properties and characterizations are studied. -P ( T ) is Definition 2.1 Let ( X, T ) be a smooth fuzzy topological space. For r I0, the family g -open set with 1 } { IX : = 0 defined by g -P ( T ) = { IX : is an r-fuzzy g }. Definition 2.2 Let ( X, T ) be a smooth fuzzy topological space. For r I0, the family g -P ( T ) satisfies the following two properties: (1) (2)
0 , 1 g -P ( T ) j g -P ( T ), for j g -P ( T ), j J. j J
g -P ( T ) ) is called a smooth g -P-fuzzy topological space. -R ( T ) is Remark 2.1 Let ( X, g -P ( T ) ) be a smooth g -P-fuzzy topological space, r I0. The family g X X defined by g -R ( T ) = { I : is r-fuzzy g -closed with 1 } { I : = 1 }. It is g -R ( T ) = { µ IX : 1 g -P ( T ) } and it satisfies the properties: straightforward that The pair ( X,
(1) (2)
0 , 1 g -R ( T ). j g -R ( T ), for j g -R ( T ), j J. j J
-P ( T ) ) be a smooth g -P-fuzzy topological space. For any IX and r I0, Definition 2.3 Let ( X, g
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A NEW VIEW ON COMPACTNESS IN SMOOTH FUZZY…
(1)
g -IP ( T ) ( , r ) = { IX : ; g -P ( T ) } is called
fuzzy interior of . (2) g -CR ( T ) ( , r ) = { IX : ;
g -R ( T ) } is called
P(T) g-
R(T) g-
fuzzy closure of . Remark 2.2 Let ( X,
g -P ( T ) ) be a smooth g -P-fuzzy topological space. For any IX and r I0, (1) If g -P ( T ) then = g -IP ( T ) ( , r ). -CR ( T ) ( , r ). (2) If g -R ( T ) then = g -P ( T ) ) and ( Y, g -P ( S ) ) be any two smooth g -P-fuzzy topological spaces Definition 2.4 Let ( X, g -P ( T ) ) ( Y, g -P ( S ) ) be a function. Then and f: ( X, g -P-fuzzy continuous function if, for each g -P ( S ), f-1 ( ) (1) f is said to be a g g -P ( T ). -P-fuzzy open function if, for each g -P ( T ), -P ( S ). f() g (2) f is called a g -P ( S ) ) be any two smooth g -P-fuzzy Proposition 2.1 Let ( X, g -P ( T ) ) and ( Y, g -P ( T ) ) ( Y, g -P ( S ) ), the f : ( X, g topological spaces, r I0. For a function, following statements are equivalent: -P-fuzzy continuous function. (a) f is a g (b)
-CR ( T ) ( , r ) ) g -CR ( S ) ( f ( ), r ). For every IX, f ( g
(c)
For every IY, f-1 ( g -CR ( S ) ( , r ) ) g -CR ( T ) ( f-1 ( ), r ).
g -IP ( S ) ( , r ) ) g -IP ( T ) ( f-1 ( ), r ). Proof: (a) (b). Let IX then, 1 ( g - CR ( S ) ( f ( ), r ) ) (d)
For every IY, f-1 (
g -P ( S ), Since f is g -
1 ( g - CR ( S )( f ( ), r ) ) g -P ( T ). Since f ( f ( ) ) f-1 ( g -CR ( S ) ( f -CR ( S ) ( f ( ), r ) ). Hence, f ( g - CR ( T ) ( , r ) ) ( ), r ) ), it follows that g -CR ( T ) ( , r ) f-1 ( g -CR ( S ) ( f ( ), r ). g (b) (c). Let IY. Then by (b), f ( g - CR ( T ) ( f-1 ( ), r ) ) g -CR ( S ) ( f ( f-1 ( ) -1 -1 -CR ( T ) ( f ( ), r ) ), r ) g -CR ( S ) ( , r ). Hence, g f ( g -CR ( S ) ( , r ) ). (c) (d). Let IY. By (c), f-1 ( g -CR ( S ) ( 1 , r ) ) g -CR ( T ) ( f-1 ( 1 ), r ) = -CR ( S ) ( 1 , r ) ) 1 - g -IP ( T ) ( f-1 ( ), r f-1 ( g g -CR ( T ) ( 1 f-1 ( ), r ) ). Then, -1
-1
P-fuzzy continuous, f (
) ).
1 - f-1 ( 1 - g g -IP ( T ) ( f-1 ( ), r ) ) 1 - f-1 ( g -CR ( S ) ( 1 , r ) ) IP ( S ) ( , r ) ) f-1 ( g -IP ( S ) ( , r ) ). - P ( S ). Then, g -IP ( S ) ( , r ) = , f-1 ( ) = f-1 ( g -IP ( S ) ( , r ) ) (d) (a). Let g IP ( T ) ( f -1 ( ), r ). Hence by definition 2.3, f-1 ( ) = g -IP ( T ) ( f -1 ( ), r ). Thus, f-1( ) g g -P ( T ). Therefore f is g -P-fuzzy continuous. -P ( T ) ) be a smooth fuzzy topological space. A g - P ( T ) covering of ( X, g -P Definition 2.5 Let ( X, g Hence,
( T ) ) is the collection { j g - P ( T ),
j J } such that 1 =
j.
j J
-P ( T ) ) is called g -P-fuzzy compact if Definition 2.6 A smooth g -P-fuzzy topological space ( X, g -P ( T ) covering of ( X, g -P ( T ) ) has a finite sub-cover. every g
g -P-fuzzy topological space ( X, g -P ( T ) ) is called g -P-fuzzy almost ( X, g -P ( T ) ), there exists a finite subset Jo of J compact if for any g -P ( T ) covering of g -CR ( T )( j, r ) = 1 , r I0. such that Definition 2.7 A smooth
j J0
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B.Amudhambigai, M.K.Uma & E.Roja
-P ( T ) ) is called g -P-fuzzy nearly compact Definition 2.8 A smooth g -P-fuzzy topological space ( X, g if for any g -P ( T ) covering of
-P ( T ) ), there exists a finite subset Jo of J such that ( X, g
g -IP ( T ) ( g -CR ( T ) ( j, r ), r ) = 1 , r I0.
j J0
-P ( T ) ) is called g -P-fuzzy S-closed if for Definition 2.9 A smooth g -P-fuzzy topological space ( X, g any
g -P ( T ) covering of ( X, g -P ( T ) ), there exists a finite subset Jo of J such that CR ( T ) ( j, r ) = j J0
1 , r I0.
g -P ( T ) ) and ( Y, g -P ( S ) ) be any two smooth g -P-fuzzy topological spaces, -P ( T ) ) ( Y, g -P ( S ) ) be a surjective and g -P-fuzzy f : ( X, g r I0. Let any function Proposition 2.2 Let ( X,
continuous function. Then for r I0, the following statements are true: (a)
-P ( S ) ). ( Y, g g -P ( T ) ) is g -P-fuzzy almost compact, then so is If ( X, g -P ( T ) )is g -P-fuzzy nearly compact, then ( Y, g -P ( S ) ) is g -P-fuzzy almost
If ( X,
(b) compact. Definition 2.10 Let ( X,
g -P ( T ) ) be a smooth g -P-fuzzy topological space and r I0. Then ( X, g -P ( -P( T ) can be written in the form of T )) is called g -P-fuzzy regular if each r-fuzzy g -open set in g X = { I : g -P( T ), g -CR ( T )( µ,r ) }. Proposition 2.3 If a smooth g -P-fuzzy topological space ( X, g -P ( T ) ) is g -P-fuzzy almost compact -P-fuzzy regular then it is g -P-fuzzy compact. and g Proposition 2.4 If a smooth g -P-fuzzy topological space ( X, g -P ( T ) ) is g -P-fuzzy nearly compact and g -P-fuzzy regular then it is g -P-fuzzy compact. -P ( S ) ) be any two smooth g -P-fuzzy Proposition 2.5 Let ( X, g -P ( T ) ) and ( Y, g -P ( T ) ) ( Y, g -P ( S ) ) be a topological spaces and any function f : ( X, g -P ( T ) ) is g -P-fuzzy almost compact then ( Y, surjective and g -P-fuzzy continuous function. If ( X, g g -P ( S ) ) is g -P-fuzzy S-closed. -P ( S ) ) be any two smooth g -P-fuzzy Proposition 2.6 Let ( X, g -P ( T ) ) and ( Y, g topological spaces and any function f : ( X, g -P ( T ) ) ( Y, g -P ( S ) ) be surjective -P-fuzzy continuous. If ( X, g -P ( T ) ) is g -P-fuzzy nearly compact then -P ( ( Y, g and g -P-fuzzy S-closed. S ) ) is g Proposition 2.7 Let ( X, g -P ( T ) ) be a smooth g -P-fuzzy topological space. Then the following statements are true: (a)
-P-fuzzy almost compact. If ( X, g -P ( T ) ) is g -P-fuzzy compact, then it is g
(b)
If ( X, g -P ( T ) ) is g -P-fuzzy compact, then it is P-fuzzy g -closed.
Proposition 2.8 Let ( X, g -P ( T ) ) be a smooth g -P-fuzzy topological space. If ( X, g -P ( T ) ) is
g -P-
g -closed as well. g -P-fuzzy nearly compact. Proposition 2.9 For every g -P-fuzzy almost compact space is -P ( S ) ) be any Definition 2.11 Let ( X, g -P ( T ) ) and ( Y, g two smooth g -P-fuzzy -P( T ) ) ( Y, g -P( S ) ) is called g -Pf : ( X, g topological spaces, r I0. Any function -1 -1 -P(S). fuzzy weakly continuous if and only if f () g -IP ( T )( f ( g -CR ( T )( j, r ) ), r ) for each g Proposition 2.10 Let ( X, g -P ( T ) ) and ( Y, g -P ( S ) ) be any two smooth g -P-fuzzy topological -P (S) ) is a g -P-fuzzy weakly continuous function and ( X, g -P ( spaces. If f : ( X, g -P ( T ) ) ( Y, g fuzzy almost compact, then it is P-fuzzy
T ) ) is a
g -P-fuzzy compact space, then (Y, S) is g -P-fuzzy almost compact, rI . 0
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A NEW VIEW ON COMPACTNESS IN SMOOTH FUZZY…
-P( T ) Definition 2.11 Let ( X, g -P ( T ) ) be a smooth g -P-fuzzy topological space. For r I0 and g -CR ( T) ( , r ) ), r ) = . is called r- g -P-fuzzy regular open set if g -IP ( T ) ( g
-P-fuzzy topological space Proposition 2.11 For any smooth g ), r I0, the following statements are equivalent :
( X,
(a)
( X, g -P ( T ) ) is g -P-fuzzy nearly compact.
(b)
For any family J of r- g -P-fuzzy regular closed sets with the property that
P( T ) and for any finite subfamily F J, it follows that
i 0 . i 0 , for any finite
iF
i = 0 . Then, 1 i = 1 , which implies that ( 1 i ) = 1 . By
i F
(a), there is a finite subfamily F of J such that contradiction. Therefore,
i 0 , i g -
iF
i F
Proof: (a) (b). Let J be a family of r- g -P-fuzzy regular closed sets such that subfamily F of J. Suppose that
g -P ( T )
i J
i J
( 1 i ) = 1 . Hence i = 0 , which is a
iF
iF
i 0 .
i J
-P-fuzzy regular open sets such that (b) (a). Let J be a family of r- g
i = 1 , i g -P( T ). Then { (
i J
1 i ) / i J } is a family of r- g -P-fuzzy regular closed sets. Suppose that for any subfamily F of J,
i 1 . Then ( 1 i ) 0 . By (b), ( 1 i ) 0 , which implies i 1 , which is a
iF
iF
i J
i J
contradiction. Therefore there exists a subfamily F of J such that
i = 1 . Hence ( X, g -P ( T ) ) is g -P-
iF
fuzzy nearly compact. 3. FUZZY g -P ( T ) FILTER AND ITS CHARACTERIZATIONS
-P ( T ) ( g -R ( T ) ) filter of a smooth In this section fuzzy g topological space is introduced and its characterizations are studied. -P ( T ) ( g -R ( T ) ) filter g -F of a smooth Definition 3.1 A fuzzy g
g -P-fuzzy
g -P-fuzzy X topological space ( X, g -P ( T ) ) is a non-empty collection of I with the following properties, r I0. -F is an r-fuzzy g -open (r-fuzzy g -open ) set in g -P ( T ) g ( g -R ( T ) ). (1) (2) 0 F, 1 F. (3) If 1, 2 F then 1 2 F. -open ) set in (4) If F and is an r-fuzzy g -open (r-fuzzy g with then F. -P ( T ) ( g -R ( T ) ) filter U on a smooth Definition 3.2 A fuzzy g
g -P ( T ) ( g -R ( T ) )
g -P-fuzzy topological -P ( T ) ( g -R ( T ) space ( X, g -P ( T ) ) is a fuzzy g -P ( T ) ( g -R ( T )) ultra filter if there is no other g ) filter finer than g -F, r I0.
-R ( T ) filter g -F in a smooth Definition 3.3 A fuzzy g
g -P-fuzzy topological space ( X, g -P ( T ) ) is called a fuzzy g -R ( T ) prime filter, if and are any two r-fuzzy g -closed sets such that
F, then F or F, r I0. Proposition 3.1 For any smooth g -P-fuzzy topological space and r I0, the following statements are equivalent : (a) (b)
g -P ( T ) ) is g -P-fuzzy compact. Every fuzzy g -R ( T ) filter F satisfies 0 . ( X,
F
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( X, g -P ( T ) )
B.Amudhambigai, M.K.Uma & E.Roja
(c)
Every fuzzy
g -R ( T ) prime filter F satisfies 0 . F
(d)
Every fuzzy g -R ( T ) ultra filter U satisfies
0.
U
Proof: (a) (b). Suppose
0 . Then, ( 1 ) 1. Since 1 is
F
r-fuzzy g -
F
open and ( X, g -P ( T ) ) is g -P-fuzzy compact, there must exists a finite sub collection {
1 2, ….., 1 n } such that 1 =
(
1 1,
1 1 ) ( 1 2 ) ….. ( 1 n ), that is, 1 2
….. n = 0 , a contradiction.
(b) (c). Follows from the fact that every fuzzy g -R ( T ) prime filter is fuzzy g -R ( T ) filter. (c) (d). The proof is obvious. (d) (a). Suppose H is a family of r-fuzzy g -closed sets with finite intersection property. For each H, consider a family G = { : is
-closed in g -R ( T ), }. Clearly, G. Let G = r-fuzzy g
{ G }. Since H has the finite intersection property, G also has the property. Thus there exists a fuzzy
H
g -R ( T ) ultra filter U such that H G U. Hence, . By hypothesis, 0 and therefore, 0 . This proves H
G
U
U
H
-P ( T ) ) is g -P-fuzzy compact. that ( X, g REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Chang. C.L : Fuzzy topological spaces, J.Math. Anal. Appl., 24(1968), 182-190. El Gayyar. M.K., Kerre. E.E. and Ramadan : Almost compactness and near compactness in smooth topological spaces, Fuzzy Sets and Systems, 62 (1994), 193-202. Lowen. R : Fuzzy topological spaces and Fuzzy compactness, J. Math. Anal . Appl., 56(1976), 621-633. Rajesh. N. and Erdal Ekici : g -locally closed sets in topological spaces, Kochi. J. Math., 2(2007), 1-9. Ramadan. A. A., Abbas S.E. and Youg Chankim : Fuzzy irresolute mappings in smooth fuzzy topological spaces, The Journal of Fuzzy Mathematics, 9(2001), 865-877. Samanta. S.K. and Chattopadhyay. K.C : Fuzzy topology : Fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Systems, 54(1993), 207-212. Smets. P :The degree of belief in a fuzzy event. Inform Sci., 25(1981), 1-19. Sostak. A.P : On a fuzzy topological structure Revid. Circ. Matem Palermo (ser II), 11(1985), 89-103. Sostak. A.P : On the neighbourhood structure of a fuzzy topological space, Zh. Rodova Univ. Nis. Ser. Math.,4(1990), 7-14. Sostak. A.P : Basic structures of fuzzy topology, J. Math. Sciences, 78(1996), 662-701. Sugeno. M : An introductory survey of fuzzy control, Inform. Sci., 36(1985), 59-83. Zadeh. L.A : Fuzzy sets, Information and Control, 8(1965), 38-353.
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