- Email: [email protected]
On fuzzy α-structure and fuzzy β-structure
ZIY
sets and systems ELSEVIER
Fuzzy Sets and Systems 92 (1997) 383-386
On fuzzy e-structure and fuzzy fi-structure M.Y. Bakier
Department of Mathematics, Facultyof Science, Universityof Assiut Assiut; Egypt Received March 1993; revisedJune 1996
Abstract
We investigate the structure of fuzzy c~-sets and fuzzy fl-sets and give some applications. @ 1997 Elsevier Science B.V.
Keywords: Fuzzy topology; Fuzzy c~-open(~-closed) sets; Fuzzy fl-open (fl-closed); Fuzzy extremely disconnected sets
1. Introduction
~-Open (~-closed) sets, fl-open (fl-closed) sets were introduced by Mashhour [5], Noiri [10] and Njastad [9]. In 1991 Singal [12] and Abdulla [1] put these concepts in fuzzy setting. In this paper the structure of these fuzzy sets and classes of fuzzy sets are investigated. Fuzzy topologies determining the same classes of fuzzy ~-sets also determine the same class of fuzzy fl-sets and vice versa. The class of fuzzy fl-sets forms a fuzzy topology if and only if the original fuzzy topology is fuzzy extremely disconnected. The class of fuzzy ~-sets always forms a fuzzy topology and fuzzy topologies generated in this way are exactly those where all fuzzy nowhere dense sets are fuzzy closed.
2. Preliminaries
Definition 2.1 (Wong [13]). L e t X be a nonempty set and I the closed unit interval [0, 1]. A fuzzy set in X is an element ofI x of all functions f r o m X to I. 0 and 1
denote the fuzzy sets given by 0(x) = 0, V x E X and l(x) = 1, VxEX. Equality of two fuzzy sets p and v on X is determined by the usual equality for mappings, i.e., # = v if #(x) = v(x), Vx E X. A fuzzy set # on X is said to be a subset of a fuzzy set v on X (written p ~
U P i I (x)=sup{t2i(x): e e l } , iEl /
VxEX.
N # i ) (x):inf{#i(x): i EI}, iEl /
VxEX.
where I denotes an arbitrary index. Definition 2.2 (Chang [3]). A fuzzy topology on a non empty set X is a collection of subsets z of 1x such that (1) 0,1Ez. (2) if # , r E z, then/~ fq vEz. (3) if #i E z, Vi EL then Ui~l ~ti E z.
0165-0114/97/$17.00 @ 1997 Elsevier ScienceB.V. All rights reserved PII S01 65-0114(96)001 71-6
384
M.Y. Bakier l Fuzzy Sets and Systems 92 (1997) 383-386
The pair (X, z) is called a fuzzy topological space. If # E z then # said to be fuzzy open set. A fuzzy set v is fuzzy closed if ve is fuzzy open. The closure of a fuzzy set # (denoted # - ) and the interior of a fuzzy set # (denoted #°) are defined by
Proof. Let the family {#i: i C I} be a family of fuzzy fl-sets then we have
#i "-~ iEl
iEl
#~
~<
#i
"
# - = n {v: v is fuzzy closed set and # ~/2}. Definition 2.3 (Pu and Liu [11]). A fuzzy set i n X is called a fuzzy point iffit takes the value 0 for all y CX except one, say, x EX. If its value atx is p (0 < p ~< 1 ) we denote this fuzzy point by o~, where the point a is called its support. Definition 2.4 (Pu and Liu [11]). The fuzzy point ap is said to be contained in a fuzzy set # or to belong to #, denoted trp E #, iff p ~<#(x). Evidently every fuzzy set can be expressed as the union of all the fuzzy points which belong to #. Definition 2.5 (Pu and Liu [11]). A fuzzy set # in (X, z) is called a neighborhood of a fuzzy point ap iff there exists a v E z such that ap C v C #, a neighborhood # is said to be open iff # is open. The family of all the neighborhoods of ap is called the system of neighborhoods of ap.
3. Fuzzy ~-strueture and fuzzy/l-structure Definition 3.1. Let z be a fuzzy topology on a nonempty set X, a fuzzy set # E I x is said to be fuzzy or-set if # ~<#°-°. The class consisting of all fuzzy a-sets is called a fuzzy or-structure and denoted by z~. Definition 3.2. Let z be a fuzzy topology on a nonempty setX, a fuzzy set # E I x is said to be fuzzy flset if # ~<#°-. The class consisting of all fuzzy fl-sets is called a fuzzy fl-structure and is denoted by z/~. We notice that if # ¢ 0 and if # is fuzzy fl-set, then #o ¢ 0 . Lenuna 3.1. A f u z z y fl-structure is closed with respect to arbitrary unions.
This completes the proof.
[]
In the following theorem we shall now characterize z~ in terms of z~.
Theorem 3.1. Let z be a f u z z y topology on X. z consists o f exactly those f u z z y sets # f o r which # N v E z~ f o r all v E z ~. Proof. Let # E z ~, v E z p, ap E # N v and let 2 be an open neighborhood of ap. Clearly 2 N #o-o too is an open neighborhood of ap, so 0 : (2 n #°-°) N v° is nonempty. Since 0 C #°- this implies 2 N (/z° n v°) = 0 n p° ¢ 0. It follows that # N vC(# ° n v°) - = (# n v) °-, i.e. # n v C z ~. Conversely, let # N v E z ~ for all v E z~. Then in particular # E z ~. Assume ap E # N (#o-o)c (pC denoting the complement of #). Clearly, ap U v E z ~ and consequently # n ( ap U v) E z #. But # n ( ap U v) = trp, hence ap is fuzzy open. As ap E #o-, this implies ap C #o-o, contrary to assumption, thus ap E # implies ap E# °-°, and # E z a. Thus we have now found that z ~ is completely determined by z ~, i.e. all fuzzy topology with the same fuzzy fl-structure also determine the same fuzzy or-structure, explicitly given by Theorem 3.1. We shall see that conversely all fuzzy topologies with the same fuzzy a-structure determine the same fuzzy fl-structure, so that z/~ is completely determined by z ~. ~]
Theorem 3.2. Every f u z z y or-structure is a f u z z y topology.
Proof. z/~ contains the empty set and is closed with respect to arbitrary unions. A standard result gives that the class of those fuzzy set # for which # N v E z #
M.Y. Bakierl Fuzzy Sets and Systems 92 (1997) 383-386
for all v E z p constitutes a fuzzy topology, hence the theorem. Henceforth, we shall also use the term fuzzy a-topology for fuzzy a-structure. Two fuzzy topologies determining the same fuzzy a-structure shall be called fuzzy a-equivalent, and the equivalence classes shall be called fuzzy a-classes. [] We may now characterize z ~ in terms o f z ~ in the following way.
Theorem 3.3. Let z be a fuzzy topology on X. Then z ~ = z al~, and hence fuzzy a-equivalent determine the same fuzzy r-structure. Proof. Let cl and int denote closure and the interior with respect to z ~. I f a p E v E z / ~ and ap E p E z ~, then #o-o M v° ~ 0 since # ° - ° is a neighborhood of %. So certainly v° meets # o - and therefore (being open) meets #°, proving # Mint(#) ~ 0, and fortiori # tO int(v) ¢ 0. This means cl(int(v)) D v, i.e. v E z ~/~. On the other hand let # E z ~/~, ap E # and ap E 2 E z. As 2 E z ~ and ap E cl(int(#)), we have, 2 fq int(#) ~ 0, and there exists a nonempty set 0 E r such that 0 c 2 n i n t ( # ) C # . In other words, 2 N #° ~ O, and ap E gO-. Thus we have verified z ~p C z ~, and the proof is complete. [] Combining Theorems 3.1 and 3.3 we get z ~ : "Ca, o r
Corollary3.1. A
fuzzy a-topology iff z = z ~.
topology
is
a fuzzy
Thus a fuzzy a-topology belongs to the fuzzy a-class o f all its determining fuzzy topologies, and is the finest fuzzy topology o f this class. Evidently z/~ is a fuzzy topology iff r ~ = z/~. In this case z ~p = z ~p = z/~, or
Corollary 3.2. I f a fuzzy r-structure B is a fuzzy topology, then B = B ~ : B p.
Theorem 3.4. The fuzzy a-sets with respect to
a
given fuzzy topology are exactly those fuzzy sets
385
which may be written as a difference between a fuzzy open set and a fuzzy nowhere dense set. Proof. If # E z ~ we have # = #o-o _ (#o-o _ #), where #o--~ _ # clearly is nowhere dense. Conversely, if # = v - 2, v E r, 2 fuzzy nowhere dense, we easily see that v C # ° - and consequently # C v C #0--% so the proof is complete. []
Corollary 3.3. A fuzzy topology is a fuzzy ortopology if and only if all fuzzy nowhere dense sets are fuzzy closed. A fuzzy a-topology can be characterized as a fuzzy topology where the difference between a fuzzy open set and a fuzzy nowhere dense is again a fuzzy open set, which is equivalent to the condition stated. T h e o r e m 3 . 5 . Fuzzy topolooies which are fuzzy
a-equivalent determine the same class of fuzzy nowhere dense sets. Proof. Let 2 be fuzzy nowhere dense set with respect to z ~. For each # E z there is a nonempty set v E r ~ such that v C # and v R )~ -- 0. As v ° ~ 0, it follows that 2 is not fuzzy dense in # with respect to z, and consequently that 2 is fuzzy nowhere dense with respect to z. Conversely, let 2 be fuzzy nowhere dense with respect to z, 2 - contains no nonempty fuzzy set from z. As cl(2) C )~- (cl denotes closure with respect to z~), cl(2) contains no nonempty fuzzy set from z ~. So 2 is fuzzy nowhere dense with respect to z ~, which completes the proof. [] Definition 3.3. A fuzzy set # is called fuzzy minimally bounded with respect to the fuzzy topology z if # C # ° - , # C # - o . Clearly this means # E z/~ and pc E z/~.
Theorem 3.6. Fuzzy topologies which are fuzzy a-equivalent determine the same class of fuzzy minimally bounded sets, and the same class of fuzzy minimally bounded open sets. Proof. The first assertion is obvious. Suppose # E z" and pC E r/~. Then since # C #0-% pC C(#C)o-, it
386
M. Y. Bakier l Fuzzy Sets and Systems 92 (1997) 383-386
follows that #o--o = # - o = / t and so # E z, and the theorem follows. [] Definition 3.4 (Ghosh [4]). A fuzzy topological (X,z) is said to be fuzzy extremally disconnected (FED, for short) if and only if the closure o f every fuzzy open set is fuzzy open in X, or equivalently, every fuzzy regularly closed set is open. Theorem 3.7. I f the fuzzy [3-structure B is a fuzzy
topology, all fuzzy topologies which z ~ = B are fuzzy extremally disconnected I f B is not a fuzzy topology, no z for which z# = B is fuzzy extremally disconnected In particular, either all or none of the fuzzy topologies of a fuzzy ~-set are fuzzy extremally disconnected Proof. Let z ~ = B, and suppose there is a # C z such that/~- ~ z. Let ap C # - - # - o . With v = ap U # - o , 2 = (#-o)e, we have ap C I~- = # - ° - C v °-, ap E 2 = ( # - o ) c ___ 2 0 - . Hence both v and 2 are in z#. The intersection v N 2 = ap is not fuzzy open since ap E p - - # - o , hence not fuzzy fl-set. So B --- z # is not a fuzzy topology. N o w suppose B is not a fuzzy topology, and z ~ = B. There is a v C z ~ such that v C. Assume v° - C z. Then v C v ° - = v°--°, contrary to the assumption. Thus we have produced a fuzzy open set whose closure is not fuzzy open, which completes the proof. []
References
[1] Abdoulla S Bin Shna, On fuzzy strong semi-continuity and fuzzy pre-continuity, Fuzzy Sets and Systems 44 (1991) 303-308. [2] N. Bourbakei, General Topology (Herman, Paris, 1966). [3] C.L. Chang, Fuzzy topological, J. Math. Anal. Appl. 24 (1968) 182-190. [4] B. Ghosh, Fuzzy extremaUy disconnected spaces, Fuzzy Sets and Systems 46 (1992) 245-250. [5] A.S. Mashhour, M.H. Ghanim and M.H. Fath, Alia, On fuzzy noncontinuous mappings, Bull. Calcutta Math. Soc. 78 (1986) 57-92. [6] M.N. Mukherjee and B. Ghosh, On fuzzy S-closed spaces and FSC sets, Bull. Malaysien Math. Soc. 12 (1989) 1-14. [7] M.N. Mukherjee and B. Ghosh, Fuzzy semi-regularization topologies and fuzzy submaximal spaces, Fuzzy Sets and Systems 44 (1991) 283-294. [8] C.V. Negoita and D.A. Rolescu, Application of Fuzzy Sets to System Analysis (Birldaauser, Basel, 1975). [91 O. Njhstad, On some classes of nearly open sets, Pacific. J. Math. 51 (1965) 961-970. [10] T. Noiri, On c~-continuousfunctions, Canopies pest. Mat. 109 (1984) 118-126. [11] P.M. Pu and Y.M. Liu, Fuzzy topology 1. Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal Appl. 76 (1980) 571-599. [12] M.K. Singal and Niti Parkas, Fuzzy preopen sets and fuzzy pre-separation axioms, Fuzzy Sets and Systems 44 (1991) 273-281. [13] C.V. Wong, Fuzzy points and local properties of fuzzy topology, J. Math. Anal Appl. 46 (1974). [14] L.A. Zadeh, Fuzzy sets, lnform, and Control 8 (1965) 338353.
sets and systems ELSEVIER
Fuzzy Sets and Systems 92 (1997) 383-386
On fuzzy e-structure and fuzzy fi-structure M.Y. Bakier
Department of Mathematics, Facultyof Science, Universityof Assiut Assiut; Egypt Received March 1993; revisedJune 1996
Abstract
We investigate the structure of fuzzy c~-sets and fuzzy fl-sets and give some applications. @ 1997 Elsevier Science B.V.
Keywords: Fuzzy topology; Fuzzy c~-open(~-closed) sets; Fuzzy fl-open (fl-closed); Fuzzy extremely disconnected sets
1. Introduction
~-Open (~-closed) sets, fl-open (fl-closed) sets were introduced by Mashhour [5], Noiri [10] and Njastad [9]. In 1991 Singal [12] and Abdulla [1] put these concepts in fuzzy setting. In this paper the structure of these fuzzy sets and classes of fuzzy sets are investigated. Fuzzy topologies determining the same classes of fuzzy ~-sets also determine the same class of fuzzy fl-sets and vice versa. The class of fuzzy fl-sets forms a fuzzy topology if and only if the original fuzzy topology is fuzzy extremely disconnected. The class of fuzzy ~-sets always forms a fuzzy topology and fuzzy topologies generated in this way are exactly those where all fuzzy nowhere dense sets are fuzzy closed.
2. Preliminaries
Definition 2.1 (Wong [13]). L e t X be a nonempty set and I the closed unit interval [0, 1]. A fuzzy set in X is an element ofI x of all functions f r o m X to I. 0 and 1
denote the fuzzy sets given by 0(x) = 0, V x E X and l(x) = 1, VxEX. Equality of two fuzzy sets p and v on X is determined by the usual equality for mappings, i.e., # = v if #(x) = v(x), Vx E X. A fuzzy set # on X is said to be a subset of a fuzzy set v on X (written p ~
U P i I (x)=sup{t2i(x): e e l } , iEl /
VxEX.
N # i ) (x):inf{#i(x): i EI}, iEl /
VxEX.
where I denotes an arbitrary index. Definition 2.2 (Chang [3]). A fuzzy topology on a non empty set X is a collection of subsets z of 1x such that (1) 0,1Ez. (2) if # , r E z, then/~ fq vEz. (3) if #i E z, Vi EL then Ui~l ~ti E z.
0165-0114/97/$17.00 @ 1997 Elsevier ScienceB.V. All rights reserved PII S01 65-0114(96)001 71-6
384
M.Y. Bakier l Fuzzy Sets and Systems 92 (1997) 383-386
The pair (X, z) is called a fuzzy topological space. If # E z then # said to be fuzzy open set. A fuzzy set v is fuzzy closed if ve is fuzzy open. The closure of a fuzzy set # (denoted # - ) and the interior of a fuzzy set # (denoted #°) are defined by
Proof. Let the family {#i: i C I} be a family of fuzzy fl-sets then we have
#i "-~ iEl
iEl
#~
~<
#i
"
# - = n {v: v is fuzzy closed set and # ~
3. Fuzzy ~-strueture and fuzzy/l-structure Definition 3.1. Let z be a fuzzy topology on a nonempty set X, a fuzzy set # E I x is said to be fuzzy or-set if # ~<#°-°. The class consisting of all fuzzy a-sets is called a fuzzy or-structure and denoted by z~. Definition 3.2. Let z be a fuzzy topology on a nonempty setX, a fuzzy set # E I x is said to be fuzzy flset if # ~<#°-. The class consisting of all fuzzy fl-sets is called a fuzzy fl-structure and is denoted by z/~. We notice that if # ¢ 0 and if # is fuzzy fl-set, then #o ¢ 0 . Lenuna 3.1. A f u z z y fl-structure is closed with respect to arbitrary unions.
This completes the proof.
[]
In the following theorem we shall now characterize z~ in terms of z~.
Theorem 3.1. Let z be a f u z z y topology on X. z consists o f exactly those f u z z y sets # f o r which # N v E z~ f o r all v E z ~. Proof. Let # E z ~, v E z p, ap E # N v and let 2 be an open neighborhood of ap. Clearly 2 N #o-o too is an open neighborhood of ap, so 0 : (2 n #°-°) N v° is nonempty. Since 0 C #°- this implies 2 N (/z° n v°) = 0 n p° ¢ 0. It follows that # N vC(# ° n v°) - = (# n v) °-, i.e. # n v C z ~. Conversely, let # N v E z ~ for all v E z~. Then in particular # E z ~. Assume ap E # N (#o-o)c (pC denoting the complement of #). Clearly, ap U v E z ~ and consequently # n ( ap U v) E z #. But # n ( ap U v) = trp, hence ap is fuzzy open. As ap E #o-, this implies ap C #o-o, contrary to assumption, thus ap E # implies ap E# °-°, and # E z a. Thus we have now found that z ~ is completely determined by z ~, i.e. all fuzzy topology with the same fuzzy fl-structure also determine the same fuzzy or-structure, explicitly given by Theorem 3.1. We shall see that conversely all fuzzy topologies with the same fuzzy a-structure determine the same fuzzy fl-structure, so that z/~ is completely determined by z ~. ~]
Theorem 3.2. Every f u z z y or-structure is a f u z z y topology.
Proof. z/~ contains the empty set and is closed with respect to arbitrary unions. A standard result gives that the class of those fuzzy set # for which # N v E z #
M.Y. Bakierl Fuzzy Sets and Systems 92 (1997) 383-386
for all v E z p constitutes a fuzzy topology, hence the theorem. Henceforth, we shall also use the term fuzzy a-topology for fuzzy a-structure. Two fuzzy topologies determining the same fuzzy a-structure shall be called fuzzy a-equivalent, and the equivalence classes shall be called fuzzy a-classes. [] We may now characterize z ~ in terms o f z ~ in the following way.
Theorem 3.3. Let z be a fuzzy topology on X. Then z ~ = z al~, and hence fuzzy a-equivalent determine the same fuzzy r-structure. Proof. Let cl and int denote closure and the interior with respect to z ~. I f a p E v E z / ~ and ap E p E z ~, then #o-o M v° ~ 0 since # ° - ° is a neighborhood of %. So certainly v° meets # o - and therefore (being open) meets #°, proving # Mint(#) ~ 0, and fortiori # tO int(v) ¢ 0. This means cl(int(v)) D v, i.e. v E z ~/~. On the other hand let # E z ~/~, ap E # and ap E 2 E z. As 2 E z ~ and ap E cl(int(#)), we have, 2 fq int(#) ~ 0, and there exists a nonempty set 0 E r such that 0 c 2 n i n t ( # ) C # . In other words, 2 N #° ~ O, and ap E gO-. Thus we have verified z ~p C z ~, and the proof is complete. [] Combining Theorems 3.1 and 3.3 we get z ~ : "Ca, o r
Corollary3.1. A
fuzzy a-topology iff z = z ~.
topology
is
a fuzzy
Thus a fuzzy a-topology belongs to the fuzzy a-class o f all its determining fuzzy topologies, and is the finest fuzzy topology o f this class. Evidently z/~ is a fuzzy topology iff r ~ = z/~. In this case z ~p = z ~p = z/~, or
Corollary 3.2. I f a fuzzy r-structure B is a fuzzy topology, then B = B ~ : B p.
Theorem 3.4. The fuzzy a-sets with respect to
a
given fuzzy topology are exactly those fuzzy sets
385
which may be written as a difference between a fuzzy open set and a fuzzy nowhere dense set. Proof. If # E z ~ we have # = #o-o _ (#o-o _ #), where #o--~ _ # clearly is nowhere dense. Conversely, if # = v - 2, v E r, 2 fuzzy nowhere dense, we easily see that v C # ° - and consequently # C v C #0--% so the proof is complete. []
Corollary 3.3. A fuzzy topology is a fuzzy ortopology if and only if all fuzzy nowhere dense sets are fuzzy closed. A fuzzy a-topology can be characterized as a fuzzy topology where the difference between a fuzzy open set and a fuzzy nowhere dense is again a fuzzy open set, which is equivalent to the condition stated. T h e o r e m 3 . 5 . Fuzzy topolooies which are fuzzy
a-equivalent determine the same class of fuzzy nowhere dense sets. Proof. Let 2 be fuzzy nowhere dense set with respect to z ~. For each # E z there is a nonempty set v E r ~ such that v C # and v R )~ -- 0. As v ° ~ 0, it follows that 2 is not fuzzy dense in # with respect to z, and consequently that 2 is fuzzy nowhere dense with respect to z. Conversely, let 2 be fuzzy nowhere dense with respect to z, 2 - contains no nonempty fuzzy set from z. As cl(2) C )~- (cl denotes closure with respect to z~), cl(2) contains no nonempty fuzzy set from z ~. So 2 is fuzzy nowhere dense with respect to z ~, which completes the proof. [] Definition 3.3. A fuzzy set # is called fuzzy minimally bounded with respect to the fuzzy topology z if # C # ° - , # C # - o . Clearly this means # E z/~ and pc E z/~.
Theorem 3.6. Fuzzy topologies which are fuzzy a-equivalent determine the same class of fuzzy minimally bounded sets, and the same class of fuzzy minimally bounded open sets. Proof. The first assertion is obvious. Suppose # E z" and pC E r/~. Then since # C #0-% pC C(#C)o-, it
386
M. Y. Bakier l Fuzzy Sets and Systems 92 (1997) 383-386
follows that #o--o = # - o = / t and so # E z, and the theorem follows. [] Definition 3.4 (Ghosh [4]). A fuzzy topological (X,z) is said to be fuzzy extremally disconnected (FED, for short) if and only if the closure o f every fuzzy open set is fuzzy open in X, or equivalently, every fuzzy regularly closed set is open. Theorem 3.7. I f the fuzzy [3-structure B is a fuzzy
topology, all fuzzy topologies which z ~ = B are fuzzy extremally disconnected I f B is not a fuzzy topology, no z for which z# = B is fuzzy extremally disconnected In particular, either all or none of the fuzzy topologies of a fuzzy ~-set are fuzzy extremally disconnected Proof. Let z ~ = B, and suppose there is a # C z such that/~- ~ z. Let ap C # - - # - o . With v = ap U # - o , 2 = (#-o)e, we have ap C I~- = # - ° - C v °-, ap E 2 = ( # - o ) c ___ 2 0 - . Hence both v and 2 are in z#. The intersection v N 2 = ap is not fuzzy open since ap E p - - # - o , hence not fuzzy fl-set. So B --- z # is not a fuzzy topology. N o w suppose B is not a fuzzy topology, and z ~ = B. There is a v C z ~ such that v C. Assume v° - C z. Then v C v ° - = v°--°, contrary to the assumption. Thus we have produced a fuzzy open set whose closure is not fuzzy open, which completes the proof. []
References
[1] Abdoulla S Bin Shna, On fuzzy strong semi-continuity and fuzzy pre-continuity, Fuzzy Sets and Systems 44 (1991) 303-308. [2] N. Bourbakei, General Topology (Herman, Paris, 1966). [3] C.L. Chang, Fuzzy topological, J. Math. Anal. Appl. 24 (1968) 182-190. [4] B. Ghosh, Fuzzy extremaUy disconnected spaces, Fuzzy Sets and Systems 46 (1992) 245-250. [5] A.S. Mashhour, M.H. Ghanim and M.H. Fath, Alia, On fuzzy noncontinuous mappings, Bull. Calcutta Math. Soc. 78 (1986) 57-92. [6] M.N. Mukherjee and B. Ghosh, On fuzzy S-closed spaces and FSC sets, Bull. Malaysien Math. Soc. 12 (1989) 1-14. [7] M.N. Mukherjee and B. Ghosh, Fuzzy semi-regularization topologies and fuzzy submaximal spaces, Fuzzy Sets and Systems 44 (1991) 283-294. [8] C.V. Negoita and D.A. Rolescu, Application of Fuzzy Sets to System Analysis (Birldaauser, Basel, 1975). [91 O. Njhstad, On some classes of nearly open sets, Pacific. J. Math. 51 (1965) 961-970. [10] T. Noiri, On c~-continuousfunctions, Canopies pest. Mat. 109 (1984) 118-126. [11] P.M. Pu and Y.M. Liu, Fuzzy topology 1. Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal Appl. 76 (1980) 571-599. [12] M.K. Singal and Niti Parkas, Fuzzy preopen sets and fuzzy pre-separation axioms, Fuzzy Sets and Systems 44 (1991) 273-281. [13] C.V. Wong, Fuzzy points and local properties of fuzzy topology, J. Math. Anal Appl. 46 (1974). [14] L.A. Zadeh, Fuzzy sets, lnform, and Control 8 (1965) 338353.