Songklanakarin J. Sci. Technol. 39 (5), 619-624, Sep - Oct. 2017

http://www.sjst.psu.ac.th

Original Article

Some results on fuzzy hyperconnected spaces

Jayasree Chakraborty*, Baby Bhattacharya, and Arnab Paul

Department of Mathematics, NIT Agartala, Tripura, 799046 India

Received: 29 April 2016; Revised: 8 August 2016; Accepted: 15 August 2016

Abstract

This paper is a prolongation of the study of fuzzy hyperconnectedness. After studying the characteristic of fuzzy hyperconnectedness concerned with different concepts we ascertain an extension of one essential result of Park et al. (2003). In particular, we obtain one result based on fuzzy hyperconnectedness which points out a comparison between the fuzzy topological spaces and the general topological spaces. A necessary and sufficient condition for a fuzzy semicontinuous function to be a fuzzy almost is established while the said two continuities are independent to each other shown by Azad (1981).

Keywords: fuzzy hyperconnected space, fuzzy regular , fuzzy nowhere , fuzzy semicontinuity, fuzzy almost continuity

1. Introduction 2. Preliminariesd

The concept of hyperconnectedness in topological Throughout this paper, simply by X and Y we shall spaces has been introduced by Steen and Seebach (1978). denote fts’s (,)X  and (,)Y  respectively and f : X Y Then Ajmal et al. (1992) have studied some of the charac- will mean that f is a function from a fts (,)X  to another fts (,)Y  . For a fuzzy set of X, cl  , int() and 1   terizations and basic properties of hyperconnected space.    X Thereafter several authors have devoted their work to (or  c ) will denote the of  , the of  and the investigate the various properties of hyperconnectedness in complement of  respectively, whereas the constant fuzzy sets taking on the values 0 and 1 on are denoted by 0 and general . Later, Caldas et al. (2002) have only defined X 1 respectively.. fuzzy hyperconnectedness in fuzzy (in X short, fts) for the exploration of the features of the so called fuzzy weakly semi-open function. 2.1 Definition (Thangaraj & Balasubramanian, 2003) A fuzzy The main seek of this paper is to study the basic set  in a fuzzy topological space (,)X  is called a fuzzy properties and preservations of fuzzy hyperconnectedness in dense if there exists no fuzzy  in (,)X  such a fts. Lastly, in this paper we define fuzzy feebly continuous that    1 . function and constitute its characterizations based on fuzzy hyperconnectedness along with other existing functions 2.2 Definition (Thangaraj & Balasubramanian, 2003) A fuzzy namely fuzzy semicontinuous function and fuzzy almost set  in a fuzzy topological space (,)X  is called a fuzzy continuous function. nowhere dense if there exists no non zero fuzzy open set  in (,)X  such that  cl()  i.e. int cl  0 . X

2.3 Definition (Azad, 1981) A fuzzy set  in a fuzzy topolo- * Corresponding author. gical space (,)X  is called fuzzy semiopen (resp. fuzzy Email address: [email protected] semiclosed) set on X if  clint  (resp. int cl   ). 620 J. Chakraborty et al. / Songklanakarin J. Sci. Technol. 39 (5), 619-624, 2017

2.4 Definition (Azad, 1981) A fuzzy set in a fuzzy topolo- if 1 is a fuzzy semi-open set of X, for each fuzzy   .  f () Y gical space (,)X  is called fuzzy regular open (resp. fuzzy regular closed) set on X if  int cl  (resp.  cl int  ). 2.16 Definition (Ekici & Kerre, 2006) A function f :  XY,(,)   is called fuzzy contra continuous if 2.5 Definition (Bin Shahna, 1991) A fuzzy set  in a fuzzy f 1 ()  for every fuzzy closed  in Y. topological space (,)X  is called fuzzy preopen (resp. fuzzy f : ( X , ) ( Y ,  ) preclosed ) set on X if  int cl  (resp. cl int   ). 2.17 Definition (Azad, 1981) A function XY from a fts X into a fts Y is said to be fuzzy almost continuous 2.6 Definition (Yalvac, 1988) Let   X be a fuzzy set and mapping if f1 ( )   X for each fuzzy regular open set  defined the following set of Y. sint   {  :    ,  is fuzzy semiopen set}, also sint   int    . 2.18 Definition (Park & Park, 2003) A function f : XY is called a regular generalized fuzzy continuous (in short, 2.7 Definition (Balasubramanian et al., 1997) A fuzzy set  rgf-continuous) if the inverse image of every fuzzy closed set in a fuzzy topological space (,)X  is called a generalized in Y is rgf-closed in X. fuzzy closed (in short, gfc)  cl( )  , whenever   and  is fuzzy open. 3. Characterizations of Fuzzy Hyperconnectedness

2.8 Definition (Park & Park, 2003) A fuzzy set  in a fts X In this section, we establish some equivalent forms is called regular generalized fuzzy closed (in short, rgf-closed) of fuzzy hyperconnectedness as a natural offshoot of the if cl()  , whenever   and  is fuzzy regular open context. Here our endeavor is to extend one particular result in X. by Azad (1981), which is imposed in his paper. Here we also find a relation in between fuzzy preclosed set and generalized 2.9 Definition (Setupathy et al., 1977) A fuzzy topological fuzzy closed set underneath of fuzzy hyperconnectedness. space (,)X  is said to be fuzzy connected if X cannot be represented as the union of two non empty disjoint open fuzzy 3.1 Definition (Caladas et al., 2002) A fts (,)X  is said to be sets on . fuzzy hyperconnected if every non-null fuzzy open subset of (,)X  is fuzzy dense in (,)X  . 2.10 Definition (Mukherjee & Ghosh, 1989) An fts (,)X  is said to be fuzzy extremally disconnected (FED, for short) iff 3.2 Example closure of every fuzzy open set is fuzzy open in X; equiva- Let X  a,, b c,   {0XX ,1,a ,0.5,  b ,0.7 , c ,0.7 , lently, every fuzzy regular closed set is fuzzy open. {a , 0.4 , b , 0.6 , c , 0.5 }}. Here every fuzzy open subset is fuzzy dense set. Therefore the fts (X ,  ) is fuzzy hyper- 2.11 Definition (Thangaraj & Soundararajan, 2013). A fuzzy . topological space (XT , ) is called a fuzzy Volterra space if cl N   1, where  ’s are fuzzy dense and fuzzy G - 3.3 Remark Every fuzzy hyperconnected space is fuzzy  i1 i  i  sets in (XT , ). connected. However the reverse of the above remark may not be true in general as shown in the following example. 2.12 Definition (Thangaraj & Poongothai, 2013). Let (XT , ) be a fuzzy topological space. Then (XT , ) is called a fuzzy 3.4 Example  X a,, b c   {0 ,1,a ,0.5, b ,0.3, c ,0.2 , -Baire space if int   0, where  ’s are fuzzy - Let  , XX        i1 i  i nowhere dense sets in (XT , ) . {a , 0.4 , b , 0.2 , c , 0.1 }} . Then the fts (X ,  ) is fuzzy connected space but not fuzzy hyperconnected space since 2.13 Definition (Thangaraj & Balasubramanian, 2001) A fuzzy every nonempty fuzzy open set is not fuzzy dense set. topological space (XT , ) is called a fuzzy P-space if count- able intersection of fuzzy open sets in (XT , ) is fuzzy open. 3.5 Theorem Every fuzzy hyperconnected space (X ,  ) is That is, every non-zero fuzzy G -set in (XT , ) is fuzzy open fuzzy extremely disconnected.  in (XT , ). Proof. Let us suppose that (X ,  ) is fuzzy hyperconnected space. Then for any fuzzy open set , cl   1 , which 2.14 Definition (Arya & Deb, 1973) A function f: X Y is   X said to be feebly continuous if, for every non empty open set implies that cl    is fuzzy open and as a consequence the V of Y, f1  V    implies that int f1  V    . space (X ,  ) is fuzzy extremally disconnected.

f : ( X , ) ( Y ,  ) 3.6 Remark The converse of the above theorem may not be 2.15 Definition (Azad, 1981) A function XY from a fts X into a fts Y is said to be fuzzy semi-continuous true in general. J. Chakraborty et al. / Songklanakarin J. Sci. Technol. 39 (5), 619-624, 2017 621

3.7 Example Proof. (i)  (ii) Let (X ,  ) be a fuzzy hyperconnected space. Let ,   {0 ,1 ,{(a , 0.5), ( b , 0.3), ( c , 0.4)}, Suppose that  is a nonempty fuzzy regular open set. Wee X  a,, b c XX  intcl(  ). (intcl )c  cl 1  cl  {(a ,0.5),( b ,0.7),( c ,0.6)},  a ,0.3 , b ,0.7 , c ,0.2 , have This implies that    X   {}a,0.5 , b ,0.7 , c ,0.4 ,{a ,0.3 , b ,0.3 ,( c ,0.2)}}.  c  1 since   0 . But this is a contradiction to the     X X One can easily verify that the fts (X ,  ) is fuzzy extremely assumption. Thus, 1 and 0 are the only fuzzy regular open X X disconnected but not fuzzy hyperconnected as the closure of sets in X. (ii) (i) Let 1 and 0 are the only fuzzy regular  X X every fuzzy open set is fuzzy open but not fuzzy dense set. open sets in X. If possible suppose that X is not fuzzy hyper- connected. This implies that there exist a nonempty fuzzy 3.8 Theorem In any fts (X ,  ), the following conditions open subset  of X such that cl   1 . We have   X are equivalent: (i) (X ,  ) is fuzzy hyperconnected space. (ii) cl int   1 . This implies that the only possibility is,    X Every fuzzy subset of (X ,  ) is either fuzzy dense or fuzzy cl int   0 . Therefore we have cl( ) 0 where    X X nowhere dense set there in.   0 which is absurd. And hence the fts (X ,  ) is a fuzzy X hyperconnected space. Proof. (i)  (ii) Let (X ,  ) be a fuzzy hyperconnected space and  be any fuzzy subset such that   1 . Suppose that  3.13 Proposition In a fuzzy hyperconnected space (X ,  ), X is not fuzzy nowhere dense. Then we have cl1 / cl (λ)  any fuzzy subset  of X is fuzzy semiopen set if int( ) 0 .  X  X 1 /int ( cl (λ)) 1 since int(( cl λ)) 0 and this implies XX X that cl int(( cl λ)) 1 But l int(( clλ)) 1  cl λ Proof. Let X be a fuzzy hyperconnected space and  be any   X .   X  , which gives that cl λ 1 . Hence  is a fuzzy dense set. fuzzy subset of X where int( ) 0 . Therefore cl( int  )    X X   (ii) (i) Let be any nonempty fuzzy open set in X. 1 . Thus  cl ( int  ). Hence, every fuzzy subset  is a  X   Now for any nonempty fuzzy open set  , we have  intcl  fuzzy semiopen. Azad (1981) had shown that the collection which implies that  is not fuzzy nowhere dense set but of fuzzy semiopen sets falls short to form the structure of using the given hypothesis, we have  is fuzzy dense set; fuzzy topology due to lacking of finite intersection property hence, the proof. Steen and Seebach (1970) have shown that of the same in a fts (,)X  . But we claim that the collection of any topological space (XT , ) is hyperconnected iff every pair fuzzy semiopen sets forms a fuzzy topology in fuzzy hyper- of nonempty open sets have non empty intersection. But the connected space, which is justified in the next proposition. above equivalent condition is partially true in the context of a fts which is demonstrated as follows: 3.14 Proposition In a fuzzy hyperconnected space (,)X  , finite intersection of fuzzy semiopen sets is fuzzy semiopen 3.9 Proposition In any fuzzy hyperconnected space (X ,  ), set. every pair of nonempty fuzzy open subset has a nonempty intersection. Proof. Let us consider that  and  are two nonempty fuzzy semiopen sets in a fuzzy hyperconnected space (X , ). So Proof. Suppose that     0 , for any two nonempty fuzzy  cl() int  and  cl() int  . Therefore cl   X       open subsets  and  of (X ,  ) . This implies that cl( int  ) 1 and cl cl() int   1 . Since and   X     X  cl λ   0 and is not a fuzzy dense set. Since  is  are two nonempty fuzzy semiopen sets in a fuzzy   X  fuzzy open, then 0  int ( cl (λ)) and  is not a fuzzy hyperconnected space and so     0 . Consequently,, X X dense set. This is a contradiction. Thus,     0 , for every clint(   )  clint   clint (  )  1 . Therefore X        X nonempty fuzzy open subsets  and  of (X ,  ) . The result    cl int   (cl int  )  cl int ( )     . It implies stated above is only the necessary condition to be a fuzzy    is fuzzy semiopen set. hyperconnected space but not sufficient which is verified in the following example. 3.15 Proposition Every fuzzy nowhere dense set is general- ized fuzzy closed set in a fts (,)X  . 3.10 Example Let X  a,, b {0 ,1 ,  ,  ,    ,    } XX 1 2 1 2 1 2 Proof. Let  be any fuzzy nowhere dense set in a fts (,)X  . where a,0.5,  b ,0.3 ,    a ,0.4,  b ,0.7. Here 1 2 Therefore int cl λ 0 and there does not exist any fuzzy   a, 0.4 , b , 0.3 }  0 but fts (X ,  ) is not fuzzy   X 1 2     X hyperconnected. Normally in fuzzy topological space the open set in between  and cl() . Also let us suppose that  following equality does not hold.   for any fuzzy open set and then we have always cl() . Therefore  is generalized fuzzy closed set. 3.11 Remark In a fuzzy hyperconnected space (X ,  ) , if  3.16 Theorem Let (,)X  be a fts. Then the following and  are two fuzzy open sets then cl   cl()()   cl  . conditions are equivalent: (i) (,)X  is fuzzy hyperconnected, 3.12 Theorem Let (X ,  ) be a fuzzy topological space. Then (ii) Every fuzzy preopen set is fuzzy dense set. the following properties are equivalent: (i) (X ,  ) is fuzzy hyperconnected, (ii) 1 and 0 are the only fuzzy regular Proof. (i)  (ii) Let us suppose that  is any fuzzy preopen X X open sets in X. set. This implies that  int cl (  ). As a result from (i) we 622 J. Chakraborty et al. / Songklanakarin J. Sci. Technol. 39 (5), 619-624, 2017 have cl cl int cl   1 . Therefore is fuzzy 4.3 Theorem Every fuzzy semi continuous function is fuzzy       X  dense set. (ii)  (i) Let  be any fuzzy preopen set. So feebly continuous in a fts (,)X  .  int cl (  ). Consequently, (ii) we give that  is fuzzy dense set. Therefore, cl  cl int cl   1 . It means that Proof. Let f : ( X ,  )  ( Y ,  ) be a fuzzy semi continuous       X (,)X  , is fuzzy hyperconnected. function. Suppose that  is a fuzzy open subset of (Y ,  ) such that f 1 ( ) 0 . This implies that 1 is a nonempty X f () 3.17 Proposition In a fuzzy hyperconnected space every fuzzy semiopen set in (X ,  ). As a result, 0 f 1   X   fuzzy preclosed set is a generalized fuzzy closed set which is sint  f1  int ( f  1 (  )) . As a result, f : ( X ,  )  ( Y ,  ) defined by Balasubramanian et al. (1997). is fuzzy feebly continuous.

Proof. Using proposition 3.15 and theorem 3.16 it can be 4.4 Remark Converse of the above theorem may not be true straightforwardly established. always.

3.18 Proposition In a fuzzy hyperconnected space every 4.5 Example X Y  a,, b c   {0 ,1, a,0.5 , b,0.4 , c,0.3 } fuzzy subset is regular generalized fuzzy closed set. Let  , XX       σ {0 ,1 , a,0.5 , b,0.7 , c,0.7 } f : ( X ,  )  ( Y ,  ) and YY       . Here , Proof. Let   where  is fuzzy regular open in X. Then defined by f a  a , f b  b , f c  c . Then f is fuzzy cl ( )   since X is fuzzy hyperconnected space and in feebly continuous function but not fuzzy semi continuous hyperconnected space only fuzzy regular open sets are 0 function. Since 1 = {(a,0.5), X f a,0.5 , b,0.7 , c,0.7 and 1 . Therefore every fuzzy subset is regular generalized (b,0.7),(c,0.7)} is not fuzzy semiopen set in X. X fuzzy closed set. In 2014, Thangaraj et al. studied the relation between fuzzy topological P-space, fuzzy hyperconnected 4.6 Proposition Let (X ,  ) be a fuzzy hyperconnected space. space and fuzzy Volterra space in a fts. If f : ( X ,  )  ( Y ,  ) is a fuzzy feebly continuous function, then f is fuzzy semicontinuous function. 3.19 Proposition (Thangaraj & Soundararajan, 2014) If the fuzzy topological P-space (XT , ) is a fuzzy hyperconnected Proof. Let f : ( X ,  )  ( Y ,  ) is a fuzzy feebly continuous space, then (XT , ) is a fuzzy Volterra space. and (X ,  ) be a fuzzy hyperconnected space. Suppose that is a fuzzy open subset of (Y ,  ) such that f 1 ( ) 0 .  X 4. Results on Fuzzy Functions in Fuzzy Hyperconnected This implies that int( f 1 ( )) 0 . Since (X ,  ) is fuzzy X Spaces hyperconnected space, therefore int( f 1 ( )) is fuzzy semiopen set in (X ,  ) . As a result, f is fuzzy semicontinuous In this section, we consider certain mappings between function. fts’s and study their behavior when either or both the domain and codomain spaces are replaced by fuzzy hyperconnected 4.7 Proposition Every fuzzy almost continuous function is spaces from which we shall obtain significant information fuzzy contra continuous function. about those functions. Proof. Let f be any fuzzy almost continuous function from X 4.1 Definition A function f : ( X ,  )  ( Y ,  ) from a fts X to Y. From the definition of fuzzy almost continuity we have into a fts Y is said to be fuzzy feebly continuous if for every f  int cl   , where  and  are two fuzzy open sets 1 fuzzy open set  of Y, f 1 ( ) 0 implies that int ( f ( )) respectively in X and Y. It implies that f ( ) cl(  ). There- X  0 . fore f is fuzzy contra continuous function. X 4.8 Remark But fuzzy contra continuous function may not 4.2 Theorem Fuzzy feebly continuous function preserves be a fuzzy almost continuous function. fuzzy hyperconnectedness. 4.9 Example Let X Y   a,, b c, τ {0 ,1 , {(a,0.5), Proof. Let f : ( X ,  )  ( Y ,  ) be a fuzzy feebly continuous XX σ {0 ,1, a,0.5 , b,0.3 , c,0.2 } surjection and let X is fuzzy hyperconnected space. Suppose (b,0.8),(c,0.7)}} and YY       . that Y is not fuzzy hyperconnected. Then there exist two non Here we define f : ( X ,  )  ( Y ,  ) , f a  b , f b  c , empty fuzzy open sets  and  such that    0 . It f c  b. Then f is fuzzy contra continuous function 1 2 1 2 Y means that f1  f  1     0 . By the hypothesis if but not fuzzy almost continuous function. Since 1  1 1 2 X 1 f 1 a,0.5 , b,0.3 , c,0.2 = a, 0.5 , b, 0.2 , c, 0.3 f1 0XX , f   2   0 , then int  f 1   0 X and 1 1  1 is not fuzzy open set in X. int  f 2   0 X . Here f  f     0 , which 11  1 2 X implies that int  f1  int  f   2   0 X . Then X is not fuzzy hyperconnected, which is a contradiction. Thus Y is 4.10 Proposition Let f: X Y be a fuzzy contra con- fuzzy hyperconnected. tinuous function from X to Y, where Y is a fuzzy hyper- J. Chakraborty et al. / Songklanakarin J. Sci. Technol. 39 (5), 619-624, 2017 623 connected space. Then f is a fuzzy almost continuous 5. Conclusions function. In this paper we sorted out the characteristic of differ- Proof. Since f is a fuzzy contra continuous so we get ent kinds of sets namely fuzzy preopen set, fuzzy preclosed f ( ) cl(  ) , where  and  is are fuzzy open sets in X set, generalized fuzzy closed set, regular generalized fuzzy and Y respectively. Again by the hypothesis X is a fuzzy closed set etc. in the ambience of fuzzy hyperconnected hyperconnected space we have f( ) cl (  )  1  f (  ) space. In literature it is found that in a fts every fuzzy closed Y intcl( )  1 . Hence the proof. Azad (1981) proved that the set is itself a generalized fuzzy closed set and also a fuzzy Y concept of fuzzy semi continuity and fuzzy almost continuity preclosed set. But using proposition 3.15 and theorem 3.16 are independent concepts in fts’s. But in the next proposition we established a relation between generalized fuzzy closed we establish a relation between these two said notions. set and fuzzy preclosed set despite the fact that there is no relation between generalized fuzzy closed set and fuzzy pre- 4.11 Proposition Let f : ( X ,  )  ( Y ,  ) be a fuzzy semi closed set. Here we also studied the behavior of numerous continuous function from X to Y, where Y is a fuzzy hyper- functions namely fuzzy febbly continuous function, fuzzy connected space. Then f is a fuzzy almost continuous almost continuous function, fuzzy semi continuous in the light function. of fuzzy hyperconnected space. At the end of this paper we culminated one particular implication which shows that fuzzy Proof. Since f is a fuzzy semi continuous function then an semicontinuity imply fuzzy almost continuity when the co inverse image of fuzzy open set in Y is fuzzy semi open in X. domain space is restricted to a fuzzy hyperconnected space. Again Y is fuzzy hyperconnected space. It gives that 1 and Y 0 are the only fuzzy regular open sets in Y using theorem References Y 3.12. But it implies that f 1 1 1 and f 1 0 0 .  YX  YX Therefore inverse image of every fuzzy regular open set in Y Ajmal, N., & Kohli, J. K. (1992). Properties of hyperconnected is fuzzy open set in X. spaces, their mappings into hausdorff spaces and embeddings into hyperconnected spaces. Acta 4.12 Proposition Let f : ( X ,  )  ( Y ,  ) be a fuzzy almost Mathematica Hungarica, 60(1-2), 41-49. continuous function from X to Y, where X is a fuzzy hyper- Arya, S. P., & Deb, M. (1973). On mappings almost continuous connected space and int f 1   0 for any non empty in the sense of Frolík. The Mathematics Student, 41,   X fuzzy open set   . Then f is a fuzzy semi continuous. 311-321. Azad, K. K. (1981). On fuzzy semi continuity, fuzzy almost Proof. Since f is a fuzzy almost continuous function, so continuity and fuzzy weakly continuity. Journal of inverse image of fuzzy regular open set in Y is fuzzy open in Mathematical Analysis and Applications, 82, 14-32. X. Again by the hypothesis, X is fuzzy hyperconnected space Balasubramanian, G., & Sundaram, P. (1997). On some with int f 1   0 for any non empty fuzzy open set generalizations of fuzzy continuous functions. Fuzzy   X   . It means that inverse image of every nonempty fuzzy Sets and Systems, 86, 93-100. open set is fuzzy semiopen set in X. Since cl int f 1    Bin Shahna, A. S. (1991). On fuzzy strong semi-continuity and 1 int f 1  . On the other hand if any fuzzy open set fuzzy pre-continuity. Fuzzy Sets and Systems, 44, 14- X     0 , then f 1   0 . Which gives that f 1  is fuzzy 32. Y   X   semiopen set in X. Thus f is a fuzzy semi continuous. Caladas, M., Navalgi, G., & Saraf, R. (2002). On fuzzy weakly semiopen functions. Proyeciones, 21, 51-63. 4.13 Remark Let f : ( X ,  )  ( Y ,  ) be any fuzzy function. Ekici, E., & Kerre, E. (2006). On fuzzy contra-continuities. Then f is fuzzy semicontinuous function iff f is fuzzy almost Advances in Fuzzy Mathematics, 1, 35-44. continuous with int f 1   0 . Thangaraj, G., & Balasubramanian, G. (2001). On fuzzy   X basically disconnected spaces. Journal of Fuzzy 4.14 Proposition If X is fuzzy hyperconnected space then Mathematics, 9(1), 103-110. any fuzzy continuous function f: X Y is regular general- Thangaraj, G., & Balasubramanian, G. (2003). On somewhat ized fuzzy continuous function. fuzzy continuous functions. Journal of Fuzzy Mathematics, 11(2), 725-736. Proof. Let f: X Y is a fuzzy continuous function and X Thangaraj, G., & Poongothai, E. (2013). On fuzzy -Baire be a fuzzy hyperconnected space. Suppose that  is any spaces. International Journal of Fuzzy Mathematics fuzzy open subset of Y. Then f 1   is regular generalized and Systems, 3(4), 275-283. fuzzy closed set in X. It implies that f is regular generalized Thangaraj, G., & Soundararajan, S. (2014). A note on fuzzy fuzzy continuous function. Volterra spaces. Annals of Fuzzy Mathematics and Informatics, 8(4), 505-510. 624 J. Chakraborty et al. / Songklanakarin J. Sci. Technol. 39 (5), 619-624, 2017

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