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SOOCHOW JOURNAL OF MATHEMATICS Volume 22, No. 1, pp. 17-32, January 1996 SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS BY SUNDER LAL AND PUSHPENDRA SINGH Abstract. In this paper some stronger forms of fuzzy continuous mappings have been introduced and some of their preservation properties are discussed. 1. Introduction In this paper we introduce and study some stronger forms of fuzzy continuous mappings in fuzzy topological spaces. These are all generalizations of mappings that already exist in general topology. In topology we come across with a large number of dierent types of sets. Using these sets we may dene a lot of mappings. It may not be pleasant task to nd adjectives for all these mappings. It would be still more dicult to nd adjectives which would reect the true nature of the mappings as also its relationship with other similar mappings. It is, therefore, proposed in 8] a more suggestive terminology. The names suggested under this scheeme may appear to be little crude for established names like continuous mappings, but are fairly appropriate for the names not so established and also for the mappings which are yet to be dened. Let and be set adjectives (or their abbreviations) associated with the topologies of X and Y . Then we say that a mapping f : X ! Y is I ( ) if the inverse image f ;1(U ) is an set in X for each set U in Y . For example, f is I (open, open) or I (o o) (in short), if f ;1(U ) is an open set in X for each open set U in Y . When X and Y are fuzzy topological spaces, then we say Received October 26, 1993. 17 18 SUNDER LAL AND PUSHPENDRA SINGH that f : X ! Y is fuzzy I ( ) if f ;1(U ) is fuzzy set of X whenever U is a fuzzy set of Y . Thus f : X ! Y is fuzzy I (o o) if f ;1(U ) is fuzzy open set of X whenever U is fuzzy open set of Y . Here we study fuzzy I ( ) mappings for some combinations of and where 2 f open(o), regular open (ro), clopen (clo), -open (po), -open (do) g and 2 f semi open (so), open (o) g. We study their interrelationships as also some of their preservation properties. 2. Preliminaries Throughout this paper X and Y mean fuzzy topological spacs. I denotes the closed unit interval. The denitions of fuzzy sets, fuzzy topological spaces and other concepts about functions can be found in 3, 12, 13]. Let U be a fuzzy set of X . U is said to be fuzzy semiopen set of X if there exists a fuzzy open set V such that V U cl V . Complement of a fuzzy semi open set is called fuzzy semi cloed. FSO(X ) denotes the family of all fuzzy semi open sets of X . S -int U (semi interior of U ) is dened as the suprimum of all fuzzy semi open sets contained in U and S -cl U (semi closure of U ) as the inmum of all fuzzy semi closed sets containing U . A fuzzy set U of a fuzzy space X is called a fuzzy regular open set of X if int cl U = U , and a fuzzy regular closed set of X if cl int U = U . A fuzzy set U of a fuzzy space X is called a fuzzy -open set of X if it is a nite union of fuzzy regular open sets of X . Complement of a fuzzy -open set is called fuzzy -closed set. A fuzzy space X is said to be fuzzy S-closed (resp: fuzzy strongly Sclosed), if every fuzzy semiopen cover of X has a nite subfamily the closures (resp: semi closures) of whose members cover X . A fuzzy space X is said to be (i) fuzzy compact 3] if every fuzzy open cover of X has a nite subcover, (ii) fuzzy nearly compact 5] if every fuzzy regular open cover of X has a nite subcover, (iii) fuzzy slightly compact, if every fuzzy clopen cover of X has a nite subcover, (iv) fuzzy almost compact 5] if every fuzzy open cover of X has a nite subfamily the closures SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 19 of whose members cover X , (v) fuzzy semi compact if every fuzzy semi open cover of X has a nite subcover. A fuzzy space X is said to be (a) fuzzy semi T0 , if every fuzzy set U of X can be written in the form U = Wi2I Vj2J Uij , where Uij are fuzzy semi open or fuzzy semi closed sets (b) fuzzy semi T1 , if every fuzzy set U of X W can be written in the form U = i2I Ui , where Ui are fuzzy semi closed sets (c) fuzzy semi T2 , if every fuzzy set U of X can be written in the form i U= _^ i2I j 2Ji Uij = _ ^ S-cl U i2I j 2Ji ij where Uij are fuzzy semi open sets (d) fuzzy S-regular, if every fuzzy open W set U can be written in the form U = i2I Ui , where Ui are fuzzy semi open sets with S -cl Ui U (e) fuzzy S-normal, if for any fuzzy closed set K and fuzzy open set U such that K U , there exists a fuzzy set V such that K S -int V S -cl V U (f) fuzzy rT0, if every fuzzy set U of X can be W V written in the form U = i2I j2J Uij , where Uij are fuzzy regular open or fuzzy regular closed sets of X (g) fuzzy rT1 , if every fuzzy set U of X can W be written in the form U = i2I Ui , where Ui are fuzzy regular closed sets of X (h) fuzzy quasi-normal, if for any fuzzy -closed set K and fuzzy -open set U such that K U , there exists a fuzzy set V such that K int V cl V U (i) fuzzy ultra-normal, if for any fuzzy closed set A and fuzzy open set B such that A B , there exists a fuzzy clopen set U such that A U B (j) Completely ultra-normal, if for any fuzzy set K and U such that cl K U and K int U there exists a fuzzy clopen set V such that K V U. i 3. Fuzzy I (o so) Mappings Denition 3.1. A mapping f : X ! Y from a fuzzy space X to fuzzy space Y is said to be fuzzy I (o so) if f ;1 (U ) is fuzzy open set of X for every fuzzy semi open set U of Y . Remark 3.2. I (o so) mappings have been studied in topological spaces 20 SUNDER LAL AND PUSHPENDRA SINGH under the name S-continuous mappings by Singal and Yadav 9]. Here we generalize this concept to the fuzzy setting. Theorem 3.3. For f : X ! Y , where X and Y are fuzzy spaces, the following statements are equivalent: (a) f is fuzzy I (o so) (b) for every fuzzy semi closed set V of Y , f ;1(V ) is fuzzy closed set of X (c) f (clU ) S -cl (f (U )), for every fuzzy set U of X (d) cl f ;1(V ) f ;1 (S -cl V ), for every fuzzy set V of Y , (e) f ;1 (S -int V ) int f ;1 (V ), for every fuzzy set V of Y . Chang 3] dened a fuzzy continuous mapping, which is fuzzy I (o o) in our terminology. Azad 1] studied fuzzy semi continuous mappings which are fuzzy I (so o) in our notations. Fuzzy irresolute mappings of Mukherjee and Sinha 7] are fuzzy I (so so) mappings. Among these mappings the following implications hold. Fuzzy I (o o) & Fuzzy I (so so) % Fuzzy I (so o) The following example shows that a mapping may be fuzzy I (o o) as well as fuzzy I (so so) but may fail to be fuzzy I (o so). Example 3.4. Let U1 , U2, V1 and V2 be fuzzy sets of I dened as follows: for each x 2 I 8> x 0 x 21 <3 U1(x) = > 2 ; 2x 12 x 34 : 3 ( 20x 0 x4 x1 1 2 U2(x) = 1 1 2 x 1 x V1(x) = 2 0 x 1 V2(x) = x 0 x 1: SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 21 Consider the fuzzy topologies 1 = f0 V1 V2 1g and 2 = f0 U1 U2 1g on I and the mapping f : (I 1 ) ! (I 2 ) dened by setting f (x) = x2 , for each x 2 I . It is easy to see that f ;1(0) = 0, f ;1(1) = 1, f ;1(U1 ) = V1, and f ;1(U2) = V2 . Thus f is fuzzy I (o o). cl U1 = U10 and hence U10 is fuzzy semi open in (I 2 ). From f ;1 (U10 ) = V10 ; cl V1 , it follows that the inverse image of fuzzy semi open set U10 in (I 2 ) is V10 which is fuzzy semiopen but not fuzzy open in (I 1 ). Hence mapping is not fuzzy I (o so). It is also easy to see that the inverse image of every fuzzy semi open set in (I 2 ) is fuzzy semi open in (I 1 ). Hence the mapping is also fuzzy I (so so). Theorem 3.5. A fuzzy I (o so) image of a fuzzy almost compact space is fuzzy strongly S-closed. Proof. Let f : X ! Y be a fuzzy I (o so) surjection and let X be fuzzy almost compact. Let fUi gi2I be a fuzzy semi open cover of Y . Since f is fuzzy I (o so). It follows that ff ;1(Ui )gi2I is a fuzzy open cover of X . Since the space is fuzzy almost compact, there exists a nite subject K of I such that _ cl f ;1(U ) = 1 i i2K X: Since f is surjective, we have f( _ cl f ;1(U )) = _ f (cl f ;1(U )) = 1 i i2K i2K i Y: Using (c) of Theorem 3.3, we get 1Y = _ f (cl f ;1(U )) _ S-cl f (f ;1(U )) = _ S-cl U : i2K i i2K i i2K i Corollary 3.6. A fuzzy I (o so) image of a fuzzy almost compact space is fuzzy S-closed. Theorem 3.7. Fuzzy I (o so) image of a fuzzy compact space is fuzzy semi compact. The proof is similar to that of Theorem 3.5. 22 SUNDER LAL AND PUSHPENDRA SINGH Corollary 3.8. Fuzzy I (o so) image of fuzzy compact space is fuzzy compact. ! Y , be a fuzzy I (o so) injection. fuzzy semi P then X is fuzzy P , where P 2 fT0 T1 T2 g 4]. Theorem 3.9. Let f : X If Y is Proof. Suppose Y is fuzzy semi T0. Let A be a fuzzy set of X . Since Y is fuzzy semi T0 , f (A) can be written in the form f (A) = Wi2I Vj2J Uij , where Uij are fuzzy semi open or fuzzy semi closed sets of Y . Since f is fuzzy I (o so) and injective, we have i A = f ;1(f (A)) = f ;1( _^U i2I j 2Ji ij ) = _ ^ f ;1(U i2I j 2Ji ij ) where f ;1(Uij ) are fuzzy open or fuzzy closed sets of X . Hence X is fuzzy T0 . Similar arguments work for P = T1 and P = T2 . Theorem 3.10. Let f : X !Y be a fuzzy I (o so) and fuzzy closed injection. Then (a) X is fuzzy regular 4] if Y is fuzzy S-regular. (b) X is fuzzy normal 4] if Y is fuzzy S-normal. Proof. (a) Let A be a fuzzy open set of X . Then f (A0) is fuzzy closed set W of Y . Y being fuzzy S-regular, we have f (A0 )0 = i2I Ui , where Ui are fuzzy open sets of Y with S -cl Ui f (A0 )0 . Since f is fuzzy I (o so) and injective, W W we have A = f ;1(f (A0 )0 ) = f ;1 ( i2I Ui ) = i2I f ;1 (Ui ), where f ;1(Ui ) are fuzzy open sets of X with cl f ;1 (Ui ) f ;1(S -cl Ui ) U . Thus X is fuzzy regular. (b) Let Y be fuzzy S-normal. Let K and U be, respectively fuzzy closed and fuzzy open sets of X , such that K U . Thus f (K ) f (U 0 )0 . Y being fuzzy S-normal, there exists a fuzzy set V of Y such that f (K ) S -int V S-cl V f (U 0)0. Since f is fuzzy I (o so) and injective, we have K = f ;1(f (K )) f ;1(S -int V ) int f ;1(V ) cl f ;1(V ) f ;1(S-cl V ) f ;1(f (U 0)0) = U: SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 23 Thus X is fuzzy normal. 4. Fuzzy I (ro so) Mappings Denition 4.1. A mapping f : X ! Y from a fuzzy space X to fuzzy space Y is said to be fuzzy I (ro so) if for every fuzzy semi open (fuzzy semi closed) set U of Y , f ;1(U ) is fuzzy regular open (fuzzy regular closed) set of X. Remark 4.2. Fuzzy I (ro so) mappig is a generalization of the concept of completely S-continuous mapping in topological spaces introduced by Singal and Yadav 11]. Obviously every fuzzy I (ro so) mapping is fuzzy I (o so) as well as I (ro o) but the converse is not true, as is shown by the following examples. Example 4.3. Let U and V be fuzzy sets of I dened as follows: for each X 2 I ( 0 if x < 13 1 if x 13 ( 0 if x < 1 2 V (x) = 1 if x 12 . U (x) = Consider fuzzy topologies 1 = f0 1g _ fW : U W 1g and 2 = f0 U 1g on I and the mapping f : (I 1) ! (I 2) dened by f (x) = x for each x 2 I . It is easy to see that the inverse image of every fuzzy semi open set in (I 2 ) is fuzzy open in (I 1 ) and hence f is fuzzy I (o so). Since U V cl U = 1, therefore V is fuzzy semi open in (I 2 ). Because int cl f ;1(V ) = 1 6= f ;1 (V ), f is not fuzzy I (ro so). Example 4.4. Let X = fx y zg, and U1 , U2 and U3 be fuzzy sets of X dened as follows: U1(x) = 0 U2(x) = 0:2 U3(x) = 0:3 U1(y) = 0:2 U2(y) = 0:5 U3(y) = 0:5 U1 (z) = 0:4 U2 (z) = 0:4 U3 (z) = 0:6: 24 SUNDER LAL AND PUSHPENDRA SINGH Consider fuzzy topologies 1 = f0 U1 U2 1g and 2 = f0 U2 1g on X and identity mapping f : (X 1 ) ! (X 2 ). It is clear that f ;1 (U2 ) is fuzzy regular open in (I 1 ), where U2 is fuzzy open set of (I 2 ) and hence mapping is fuzzy I (ro o). Since inverse image of fuzzy semi open set U3 in (I 2 ) is fuzzy semi open which is not fuzzy regular open in (I 1 ), because int cl f ;1(U3 ) = U2 6= f ;1 (U3 ). Hence mapping is not fuzzy I (ro so). Remark 4.5. In general topology I (ro o) mappings have been studied under the name completely continuous 2]. Theorem 4.6. A fuzzy I (ro so) image of a fuzzy nearly compact space is fuzzy semi compact. Proof. Let f : X ! Y be a fuzzy I (ro so) mapping from a fuzzy nearly compact space X on to a fuzzy space Y . Suppose fUi gi2I be a fuzzy semi open cover of Y . Since f is fuzzy I (ro so), therefore ff ;1 (Ui )gi2I is a fuzzy regular open cover of X . Since the space X is fuzzy nearly compact therefore, W there exists a nite subset K of I such that i2K f ;1 (Ui ) = 1X . By surjectivity of f , we have f( _ f ;1(U )) = _ f (f ;1(U )) = _ U = 1 i2K i i2K i i2K i Y: Thus Y is fuzzy semi compact. Theorem 4.7. Let f : X ! Y be a fuzzy I (ro so) injection from a fuzzy space X to fuzzy space Y . If Y is fuzzy semi T0 , then X is fuzzy rT0 . Proof. Let A be a fuzzy set of X . Then f (A) being a fuzzy set of a fuzzy W V semi T0 space Y , can be written in the form f (A) = i2I j2J Uij , where Uij are fuzzy semi open or fuzzy semi closed sets of Y . Since f is fuzzy I (ro so) injection we have A = f ;1(f (A)) = f ;1( _^U i2I j 2Ji ij ) = _ ^ f ;1(U i2I j 2Ji i ij ) SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 25 where f ;1 (Uij ) are fuzzy regular open or fuzzy regular closed sets of X . Thus X is fuzzy rT0. Using similar argument one can prove. Theorem 4.8. Let f : X !Y fuzzy semi T1 , then X is fuzzy rT1 . be a fuzzy I (ro so) injection. If Y is 5. Fuzzy I (clo so) Mappings Denition 5.1. A mapping f : X ! Y from a fuzzy space X to fuzzy space Y is said to be fuzzy I (clo so) if the inverse image of every fuzzy semi open (or fuzzy semi closed) set of Y is fuzzy clopen set of X . Remark 5.2. Fuzzy I (clo o) mappings are a generalization of totally S-continuous mappings 11]. Obviously every fuzzy I (clo so) mapping is fuzzy I (ro so) as well as fuzzy I (clo o) mapping, but the converse need not be true. Example 5.3. Let U and V be fuzzy sets of I dened as follows: for each x 2 I ( 1 ; x 0 x 1 2 U (x) = 1 x 1 x 2 ( x 0 x 1 2 V (x ) = 1 1 ; x 2 x 1. Consider fuzzy topologies 1 = fW W 0 : 0 W < V g _ fV g and 2 = f0 U 1g and the mapping f : (I 1) ! (I 2) dened by f (x) = 1 ; x, for each x 2 I . It is clear that the mapping is fuzzy I (ro so). Since U is fuzzy open and hence fuzzy semi open in (I 2 ). f ;1(U ) = V is fuzzy regular open in (I 1 ) which is not fuzzy clopen. Thus the mapping is not fuzzy I (clo so). Example 5.4. Suppose that U1 and U2 are fuzzy sets of X as described 26 SUNDER LAL AND PUSHPENDRA SINGH in Example 4.4. Consider fuzzy topologies 1 = f0 U1 U10 U2 1g and 2 = f0 U1 1g. Let f : (X 1) ! (X 2) be the identity mapping. Clearly, inverse image of every fuzzy open set in (X 2 ) is fuzzy clopen in (X 1 ). Hence the mapping is fuzzy I (clo o). If we take inverse image of fuzzy semi open set U2 in (X 2 ), then it is fuzzy open which is not fuzzy clopen in (X 1 ). Hence the mapping is not fuzzy I (clo so). Remark 5.5. In topology I (clo o) mappings have been studied under the name totally continuous 6]. Theorem 5.6. A fuzzy I (clo so) image of a fuzzy slightly compact space is fuzzy semi compact. Proof. Let f : X ! Y be a fuzzy I (clo so) mapping from a fuzzy slightly compact space X on to fuzzy space Y . Let fUi gi2I be a fuzzy semi open cover of Y . Since f is fuzzy I (clo so), ff ;1 (Ui )gi2I is a fuzzy clopen cover of X . Since the space is fuzzy slightly compact, there exists a nite subset K of I W such that i2K f ;1(Ui ) = 1X . Since f is surjective, we have f( _ f ;1(U ) = _ f (f ;1(U )) = _ U = 1 i2K i Thus Y is fuzzy semi compact. i2K i i2K i Y: 6. Fuzzy I (clo o), Fuzzy I (ro o), Fuzzy I (po o), Fuzzy I (do o) and Fuzzy Clopen Mappings Denition 6.1. A mapping f : X ! Y from a fuzzy space X to a fuzzy space Y is said to be (a) fuzzy I (do o), if f ;1 (U ) is fuzzy -open set of X for each fuzzy open set U of Y (b) fuzzy I (po o), if f ;1(U ) is fuzzy -open set of X for each fuzzy open set U of Y (c) fuzzy I (ro o), if f ;1(U ) is fuzzy regular open set of X for each fuzzy open set U of Y SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 27 (d) fuzzy I (clo o), if f ;1 (U ) is fuzzy clopen set of X for each fuzzy open set U of Y . Among these various types of mappings the following implications hold. Fuzzy I (clo o) )Fuzzy I (ro o) ) Fuzzy I (po o) ) Fuzzy I (do o) )Fuzzy I (o o): However, none of these implications is reversible as is shown by the following examples. Example 6.2. Let U1 and U2 be fuzzy sets of I dened as follows: for every x 2 I , ( x 0 x 1 , 2 U1 (x) = 1 1 ; x 2 x 1 ( 2x 0 x 12 , U2 (x) = 2(1 ; x) 21 x 1. Consider fuzzy topology = f0 U1 U2 1g on I and the mapping f : (I ) ! (I ) dened by f (x) = x, for every x 2 I . We see that inverse image of every fuzzy open set is fuzzy open and hence f is fuzzy I (o o). Inverse image of fuzzy open set U2 is not fuzzy -open. Hence f is not fuzzy I (do o). Example 6.3. Let us consider fuzzy set fab of R dened as follows: for every x 2 R, fab = 1 = 0 if a < x < b otherwise: Consider fuzzy topology 1 on R generated by fuzzy sets ffab : a b 2 R a < bg and its relative topology 2 on the interval ;1 1]. Then mapping f : (R 1 ) ! (;1 1] 2 ) dened by f (x) = sin x is fuzzy I (do o), since 1 is fuzzy semi regular space, i.e: int cl fab = fab in (R 1 ). It is clear that fuzzy 28 SUNDER LAL AND PUSHPENDRA SINGH set g dened by setting g(x) = 1 ; 21 < x < 12 = 0 otherwise is fuzzy open in (;1 1] 2 ). We have sin;1 (g(x)) = g(sin x) i.e. ; 12 < sin x < 12 = 1 if = 0 otherwise sin;1 (g(x)) = 1 if x 2 (n ; =6 n + 6 ) = 0 otherwise = _1 nf n=;1 ab (x) : x ;1 n 1 o 2 (n ; 6 n + 6 ) which being an innite union of fuzzy regular open set is not fuzzy -open. The mapping is not fuzzy I (po o). Example 6.4. Let U1, U2 and U3 be fuzzy sets of I dened as follows: for every x 2 I , ( 0 0 x 21 , 1 8> 21x ; 1 2 0x x1 1 , 4 < 1 U2(x) = > ;4x + 2 4 x 21 , : 0 1 x 1, 2 8> 1 0 x 41 , < U3(x) = > ;4x + 2 14 x 21 , : 2x ; 1 12 x 1. Consider fuzzy topologies 1 = f0 U1 U2 U1 _ U2 1g and 2 = f0 U3 1g on I and the mapping f (I 1) ! (I 2 ) dened by f (x) = x, for each x 2 I . f ;1(U3) = U1 _ U2 , which is fuzzy -open but not fuzzy regular open in (I 1 ). Thus mapping is fuzzy I (po o) which is not fuzzy I (ro o). U1(x) = SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 29 Example 6.5. Let us take fuzzy sets U1 and U2 of I which are dened in Example 6.4. Consider fuzzy topologies 1 = f0 U1 U2 U1 _ U2 1g and 2 = f0 U1 1g on I and the mapping f : (I 1 ) ! (I 2 ) dened by f (x) = x, for every x 2 I . It is clear that inverse image of fuzzy open set U1 in (I 2 ) is fuzzy regular open in (I 1 ) which is not fuzzy clopen. Hence the mapping is fuzzy I (ro o) but not fuzzy I (clo o). Denition 6.6. A mapping f : X ! Y from a fuzzy space X to fuzzy space Y is said to be fuzzy clopen mapping if f (U ) is fuzzy clopen set of Y for each fuzzy clopen set of X . Clearly, every map which is both fuzzy open as well as fuzzy closed is a fuzzy clopen map, but the converse is not true. Example 6.7. Let U1 and U2 be fuzzy sets of I dened by setting ( x 0 x 1 , 2 U1 (x) = 1 1 ; x 2 x 1 U2 (x) = x 0 x 1: Consider fuzzy topologies 1 = f0 U1 U10 U2 1g and 2 = f0 U1 U10 1g on I and the mapping f : (I 1 ) ! (I 2 ) dened as f (x) = x, for every x 2 I . We see that image of every fuzzy clopen set in (I 1 ) is fuzzy clopen in (I 2 ), the mapping is fuzzy clopen. Image of fuzzy open set U2 in (I 1 ) is not fuzzy open in (I 2 ) and image of fuzzy closed set U20 in (I 1 ) is not fuzzy closed in (I 2 ). Thus the mapping is neither fuzzy open nor fuzzy closed. Theorem 6.8. Let f : X ! Y be a fuzzy I (o o) fuzzy clopen mapping from fuzzy space X on to fuzzy space Y . If Y is fuzzy ultra-normal or fuzzy completely ultra-normal then so is Y . Proof. Let X be a fuzzy ultra-normal and let A and B be, respectively fuzzy closed and fuzzy open sets of Y such that A B . Since f is fuzzy I (o o), f ;1(A) and f ;1(B ) are respectively fuzzy closed and fuzzy open sets of X such that f ;1(A) f ;1(B ). Since X is fuzzy ultra-normal there exists 30 SUNDER LAL AND PUSHPENDRA SINGH a fuzzy clopen set U such that f ;1 (A) U f ;1 (B ). By surjectivety of f , we have A = f (f ;1(A)) f (U ) f (f ;1(B )) = B: Since f is fuzzy clopen mapping f (U ) is fuzzy clopen set of Y . Thus Y is fuzzy ultra-normal. Next, let Y be fuzzy completely ultra-normal, and let A and B be fuzzy sets of Y such that clA B and A int B . Then f ;1 (A) and f ;1 (B ) are fuzzy sets of X such that f ;1 (clA) f ;1 (B ) and f ;1 (A) f ;1(int B ). Since f is fuzzy I (o o), f ;1(clA) is a fuzzy closed set and f ;1(int B ) is a fuzzy open set. Now f ;1(clA) is a fuzzy closed set such that f ;1(A) f ;1 (clA). Therefore cl(f ;1(A)) f ;1(clA) f ;1(B ). Similarly f ;1 (A) f ;1( int B ) int f ;1(B ). Since X is fuzzy completely ultra-normal, there exists a fuzzy clopen set V such that f ;1(A) V f ;1(B ). By surjectivity of f , we have A = f (f ;1(A)) f (V ) f (f ;1(B )) = B , with clA = f (f ;1(clA)) f (f ;1(B )) = B , and A = f (f ;1(A)) f (f ;1(int B )) =int B . Also f (V ) is fuzzy clopen set, since f is a fuzzy clopen mapping. Thus Y is fuzzy completely ultra-normal. Theorem 6.9. If f : X !Y is a fuzzy open fuzzy closed and fuzzy I (po o) surjection of a fuzzy space X to a fuzzy space Y . If X is fuzzy quasinormal, then Y is fuzzy normal. Proof. Let A be a fuzzy closed and B be a fuzzy open set of Y such that A B . Since f is fuzzy I (po o), f ;1(A) is fuzzy -closed and f ;1(B ) is fuzzy -open set of X such that f ;1(A) f ;1(B ). By fuzzy quasi-normality of X , there exists a fuzzy open set V such that f ;1(A) V clV f ;1(B ). Since f is fuzzy open and surjective, we have A = f (f ;1(A)) f (V ) f (clV ) f (f ;1(B )) = B: Also, f is fuzzy closed we have A f (V ) clf (V ) B SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 31 where f (V ) is fuzzy open. Thus Y is fuzzy normal. Theorem 6.10. If X is a fuzzy quasi-normal space and f : X ! Y is fuzzy I (po o) surjection which takes fuzzy open sets of X to fuzzy clopen sets of Y , then Y is fuzzy ultra-normal. Proof. Let A and B be, respectively fuzzy closed and fuzzy open sets of Y such that A B . Then f ;1(A) and f ;1(B ) are respectively fuzzy closed and fuzzy -open sets of X such that f ;1 (A) f ;1(B ), since f is fuzzy I (po o). By fuzzy quasi-normality of X , there exists a fuzzy open set V such that f ;1 (A) V clV f ;1 (B ). By surjectivity of f , we have A = f (f ;1(A)) f (V ) f (clV ) f (f ;1(B )) = B: We have A f (V ) B , where f (V ) is fuzzy clopen set of Y . Thus Y is fuzzy ultra-normal. References 1] K. K. Azad, Onfuzzysemicontinuityfuzzyalmostcontinuityandfuzzyweaklycontinuity, J. Maths. Anal. Appl., 82(1981), 14-32. 2] S. P. Arya and R. Gupta, Onstronglycontinuousmappings, Kyungpook Math., 14 (1974), 131-143. 3] C. L. Chang, Fuzzytopologicalspaces, J. Math. Anal. Appl., 24(1968), 182-193. 4] B. Hutton and I. Reilly, Separationaxiomsinfuzzytopologicalspaces, Fuzzy Sets and Systems, 3(1980), 93-104. 5] Es A. Haydor, Almostcompactnessandnearcompactnessinfuzzytopologicalspaces, Fuzzy Sets and Systems, 22(1987), 289-202. 6] R. C. Jain, Ph.D. Thesis, Meerut University, 1981. 7] M. N. Mukherjee and S. P. Sinha, Irresoluteandalmostopenfunctionsbetweenfuzzytopological spaces, Fuzzy Sets and Systems, 29(1989), 381-388. 8] S. Lal, Some stronger forms of continuity and normality in topological spaces. Proc. 2nd Biennial Conf. Allahabad Maths. Soc., 1990, 23-29. 9] A. R. Singal and D. S. Yadav, S -continuousfunctions, Ganita Sandesh, 2(1988), 82-86. 10] A. R. Singal and D. S. Yadav, Ageneralizationofsemicontinuousmappings, Bihar Maths. Soc., 11(1987), 1-9. 11] A. R. Singal and D. S. Yadav, Completely S-continuous and totally S-continuous mappings (to appear). 12] Tuna Hatice Yalvac, Fuzzysetsandfunctionsonfuzzyspaces, J. Math. Anal. Appl., 126(1987), 409-423. 32 SUNDER LAL AND PUSHPENDRA SINGH 13] L. A. Zadeh, Fuzzysets, Inform. and Control, 8(1965), 338-353. Department of Mathematics, Institute of Basic Science, Agra University, Khandari, Agra 282002, India.