Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 40, 2021, pp. 151-162 151 Regular generalized fuzzy b-separation axioms in fuzzy topology Varsha Joshi * Jenifer J.Karnel † Abstract Regular generalized fuzzy b-closure and regular generalized fuzzy b- interior are stated and their characteristics are examined, also Regular generalized fuzzy b -πœπ‘– separation axioms have been introduced and their interrelations are examined. The characterization of regular generalized fuzzy b –separation axioms are analyzed. Keywords: rgfbCS; rgfbOS; rgfbCl; rgfbInt; rgfbT0; rgfbT1; rgfbT2; rgfbT2 1 2 and fuzzy topological spaces X (in short fts). 2020 AMS subject classification: 54A40 * Mathematics Department, SDM College of Engineering & Technology, Dharwad-580 003. Karnataka, India.E-mail: varshajoshi2012@gmail.com * Mathematics Department, SDM College of Engineering & Technology, Dharwad-580 003. Karnataka, India.E-mail: jeniferjk17@gmail.com †Received on January 12th, 2021. Accepted on May 12th, 2021. Published on June 30th, 2021. doi: 10.23755/rm.v40i1.624. ISSN: 1592-7415. eISSN: 2282-8214. Β©The Authors.This paper is published under the CC-BY licence agreement. Varsha Joshi and Dr.Jenifer J.Karnel 152 1. Introduction The fundamental theory of fuzzy sets were introduced by Zadeh [16] and Chang [9] studied the theory of fuzzy topology. After this Ghanim.et.al [10] introduced separation axioms, regular spaces and fuzzy normal spaces in fuzzy topology. The theory of regular generalized fuzzy b-closed set (open set) presented by Jenifer et. al [11]. In this study we define rgfb-closure, rgfb- interior and rgfb-separation axioms and their implications are proved. Effectiveness nature of the various concepts of fuzzy separation ideas are carried out. Characterizations are obtained. 2. Preliminary (X1, Ο„),( X2, Οƒ) (or simply X1, X2 ) states fuzzy topological spaces(in short, fts) in this article. Definition 2.1[1, 3]: In fts X1, 𝛼 be fuzzy set. (i) If 𝛼 =IntCl(𝛼)then 𝛼 is fuzzy regular open(precisely, frOS). (ii) If 𝛼 =ClInt(𝛼)then 𝛼 is fuzzy regular closed (precisely, frCS). (iii) If ( ) ( ) ClIntIntCl οƒšο‚£ then 𝛼 is f b-open set (precisely, fbOS). (iv) If ( ) ( ) ClIntIntCl  then 𝛼 is f b-closed set (precisely, fbCS). Remark 2.2 [1]: In a fuzzy topological space X, The following implication holds good Figure1. Interrelations between some fuzzy open sets Definition 2.3[3]: Let  be a fuzzy set in a fts X1. Then , (i)   = , )fbCS(X a is : )bCl( 1 . (ii)   ο‚£οƒš= ),fbOS(X a is :)bInt( 1 . Definition 2.4[11]: In a fts X1, if bCl( ) ≀ 𝛽 , at any time when  ≀ 𝛽, then fuzzy set  is named as regular generalized fuzzy b-closed (rgfbCS).Where 𝛽 is fr- open. . Remark 2.5[11]: In a fts X1, if 1- is rgfbCS(X1 ) then fuzzy set  is rgfbOS. frOS f-open fbOS Regular generalized fuzzy b-separation axioms in fuzzy topology 153 Definition 2.6[11]: In a fts X1, if bInt(  )β‰₯ 𝛽 , at any time when  β‰₯ 𝛽, then fuzzy set  is named as regular generalized fuzzy b-open (rgfbOS).Where 𝛽 is fr- closed. Definition 2.7[13]: Let (X1, Ο„), (X2, Οƒ) be two fuzzy topological spaces. Let f : X1 β†’ X2 be mapping, (i) if f-1() is rgfbCS(X1), for each closed fuzzy set  in X2, then f is said to be regular generalized fuzzy b-continuous (briefly, rgfb- continuous). (ii) if f-1() is open fuzzy in X1, for each rgfbOS  in X2, then f is called strongly rgfb-continuous. (iii) if f-1() is rgfbCS in X1, for each rgfbCS  in X2, then f is called rgfb-irresolute. Definition 2.8[10]: X1 is a fts which is named as (i) fuzzy T0(in short, fT0) if and only if for each pair of fuzzy singletons p1 and p2 with various supports there occurs open fuzzy set U such that either p1≀ U ≀ 1- p2 or p2≀ U ≀ 1-p1 . (ii) fuzzy T1(in short fT1)if and only if for each pair of fuzzy singletons p1 and p2 with various supports, there occurs open fuzzy sets U and V such that p1≀ U≀1- p2 and p2≀ V ≀ 1- p1 . (iii) fuzzy T2(in short, fT2) or f-Hausdorff if and only if for each pair of fuzzy singletons p1and p2 with various supports ,there occurs open fuzzy sets U and V such that p1≀U≀ 1- p2 , p2≀V≀ 1- p1 and U≀ 1- V. (iv) fuzzy T2 1 2 (in short, fT2 1 2 ) or f-Urysohn if and only if for each pair of fuzzy singletons p1 and p2 with various supports, there occurs open fuzzy sets U and V such that p1≀U≀ 1- p2 , p2≀V≀ 1- p1 and clU≀ 1-cl V. 3. Regular generalized fuzzy b-closure (rgfbCl) and Regular generalized fuzzy b-Interior (rgfbInt) Definition 3.1:The regular generalized fuzzy b-closure is denoted and defined by, rgfbCl ( ) = Ξ› {  :  is a rgfbCS( X1), β‰₯  }. Where  be a fuzzy set in fts X1. Theorem 3.2:Let X1 be fts, then the properties that follows are occurs for rgfbCl of a set Varsha Joshi and Dr.Jenifer J.Karnel 154 i. rgfbCl(0) = 0 ii. rgfbCl(1) = 1 iii. rgfbCl( ) is rgfbCS in X1 iv. rgfbCl[rgfbCl( )] = rgfbCl( ) Definition 3.3:Let  and 𝛽 be fuzzy sets in fuzzy topological space X1. Then regular generalized fuzzy b-closure of (  V 𝛽) and regular generalized fuzzy b-closure of (  /\ 𝛽) are denoted and defined as follows i. rgfbCl ( V 𝛽 ) = /\ {  :  is a rgfbCS(X1) , where  β‰₯ ( V 𝛽 ) } ii. rgfbCl ( /\𝛽 ) = /\ {  :  is a rgfbCS(X1), where  β‰₯ ( /\ 𝛽) } Theorem 3.4: Let  and 𝛽 be fuzzy sets in fts X1, then the following relations occurs i. rgfbCl( ) V rgfbCl(𝛽) ≀ rgfbCl( V 𝛽) ii. rgfbCl( ) /\ rgfbCl(𝛽) β‰₯ rgfbCl( /\ 𝛽) Proof: (i) We know that  ≀ ( V 𝛽) or 𝛽 ≀ ( V 𝛽) οƒž rgfbCl ( ) ≀rgfbCl ( V 𝛽) orrgfbCl(𝛽) ≀rgfbCl ( V 𝛽) Hence, rgfbCl (  ) V rgfbCl (𝛽) ≀rgfbCl ( V 𝛽). (ii) We know that  β‰₯ ( /\𝛽) or 𝛽 β‰₯ ( /\𝛽) οƒž rgfbCl( ) β‰₯rgfbCl ( /\𝛽) orrgfbCl(𝛽) β‰₯rgfbCl ( /\𝛽) Hence, rgfbCl(  ) /\ rgfbCl(𝛽) β‰₯rgfbCl ( /\𝛽). Theorem 3.5:  is rgfbCS in a fts X1, if and only if  =rgfbCl( ). Proof: Suppose  is rgfbCS. Since  ο‚£  and  οƒŽ { 𝛽: 𝛽 is rgfbCS(X1) and  ο‚£ 𝛽 },  is the smallest and contained in 𝛽,therefore  =Ξ›{ 𝛽: 𝛽 is rgfbCS( X1 )and  ο‚£ 𝛽}=rgfbCl( ). Hence,  =rgfbCl ( ). On the other hand, Suppose  =rgfbCl ( ), then  = Ξ›{ 𝛽: 𝛽 is rgfbCS,  ο‚£ 𝛽 } οƒž  οƒŽ Ξ› { 𝛽: 𝛽 is rgfbOS,  ο‚£ 𝛽 }. Hence,  is rgfbCS. Definition 3.6: The regular generalized fuzzy b-interior is denoted and defined by, rgfbInt(  ) = V { 𝛿: 𝛿 is a rgfbOS(X1), ο‚£  }. Where  be a fuzzy set in fts X1. Theorem 3.7: Let X1 be fts, then the properties that follows are occurs for rgfbInt of a set i. rgfbInt(0) = 0 ii. rgfbInt(1) = 1 iii. rgfbInt(  ) is rgfbOS in X1 iv. rgfbInt[rgfbInt(  )] = rgfbInt( ) . Regular generalized fuzzy b-separation axioms in fuzzy topology 155 Definition 3.8: Let  and 𝛽 are fuzzy sets in fts X1. Then regular generalized fuzzy b-interior of ( V 𝛽) and regular generalized fuzzy b-interior of ( /\ 𝛽) are denoted and defined as follows i. rgfbInt (  V 𝛽 ) = V {𝛿: 𝛿 is a rgfbOS( X1), where 𝛿 ≀ ( V 𝛽 ) }. ii. rgfbInt(  /\ 𝛽 ) = V { 𝛿: 𝛿 is a rgfbOS(X1), where 𝛿 ≀ ( /\ 𝛽 )}. Theorem 3.9:Let  and 𝛽 are fuzzy sets in fts X1, then the following relations occurs i. rgfbInt(  ) V rgfbInt(𝛽) ≀ rgfbInt( V 𝛽) ii. rgfbInt(  ) /\ rgfbInt(𝛽) β‰₯ rgfbInt( /\𝛽) Proof: (i) We know that,  ≀ ( V 𝛽) or 𝛽 ≀ ( V 𝛽) οƒž rgfbInt(  ) ≀ rgfbInt ( V 𝛽) or rgfbInt(𝛽) ≀ rgfbInt ( V 𝛽) Hence, rgfbInt(  ) V rgfbInt(𝛽) ≀ rgfbInt( V 𝛽). (ii)We know that  β‰₯ ( /\𝛽) or 𝛽 β‰₯ ( /\𝛽) οƒž rgfbInt( ) β‰₯ rgfbInt ( /\𝛽) or rgfbInt(𝛽) β‰₯ rgfbInt ( /\𝛽) Hence, rgfbInt(  ) /\ rgfbInt(𝛽) β‰₯ rgfbInt ( /\𝛽). Theorem 3.10: Let X1 be fts,  is rgfbOS if and only if  =rgfbInt( ). Proof: Suppose  is rgfbOS. Since  ο‚£  ,  οƒŽ { 𝛿: 𝛿 is rgfbOS and 𝛿  } Since biggest  contains 𝛿. Therefore,  = V{ 𝛿: 𝛿 is rgfbOS 𝛿 ο‚£  } = rgfbInt (  ). Hence,  =rgfbInt( ). On the other hand, Suppose  =rgfbInt ( ).Then,  =V{ 𝛿: 𝛿 is rgfbOS, 𝛿 ο‚£  } οƒž  οƒŽ V { 𝛿: 𝛿 is rgfbOS 𝛿 ο‚£  }. Hence,  is rgfbOS. Theorem 3.11: Let  be a fuzzy set in a fts X1, in that case following relations holds good i. rgfbInt(1- ) = 1-rgfbCl( ) ii. rgfbCl(1- ) = 1- rgfbInt( ) Proof: (i) Let  be a fuzzy set in fts X1. Then we have rgfbCl ( ) = Ξ› {  :  is a rgfbCS( X1), β‰₯  }. Where  be a fuzzy set in fts X1. 1-rgfbCl (  ) = 1- Ξ› {  :  is a rgfbCS( X1), β‰₯  }. = V { 1 βˆ’  :  is a rgfbCS( X1), β‰₯  }. = V{ 1 βˆ’  : 1 βˆ’  is a rgfbOS( X1), ≀ 1- }. = rgfbInt (1- ) Hence, 1-rgfbCl(  ) = rgfbInt (1- ). (ii) Let  be a fuzzy set in fts X1. Then we have rgfbInt(  ) = V { 𝛿: 𝛿 is a rgfbOS(X1), ο‚£  }. Where  be a fuzzy set in fts X1. Varsha Joshi and Dr.Jenifer J.Karnel 156 1-rgfbInt (  ) = 1-V { 𝛿 : 𝛿 ο‚£  and 𝛿 is rgfbOS (X1)} = Ξ›{1- 𝛿: 𝛿 ο‚£  and 𝛿 is rgfbOS(X1)} = Ξ› {1- 𝛿 : 1- ο‚£ 1- 𝛿 and 1- 𝛿 is rgfbCS( X1)} = rgfbCl (1- ) Hence 1-rgfbInt (  ) = rgfbCl (1- ). 4. rgfb-separation axioms Definition 4.1:A fts is known as rgfbT0, that is regular generalized fuzzy bT0, iff for each pair of fuzzy singletons q1 and q2 with various supports, there occurs rgfbOS 𝛿 such that either q1≀ 𝛿 ≀ 1- q2 or q2≀ 𝛿 ≀ 1- q1. Theorem 4.2: A fts is rgfbT0,that is regular generalized fuzzy bT0, if and only if rgfbCl of crisp fuzzy singletons q1 and q2 with various supports are different. Proof: To prove the necessary condition: Let a fuzzy topological space be rgfbT0 and two crisp fuzzy singletons be q1 & q2 with various supports x1 & x2 respectively i.e. x1 β‰  x2. Since fts is rgfbT0 ,there exist a rgfbOS 𝛿 such that, q1≀ 𝛿 ≀ 1- q2 οƒž q2 ≀ 1- 𝛿, but q2≀ rgfbCl(q2) ≀ 1- 𝛿, where q1≀ rgfbCl(q2) οƒž q1≀ 1- 𝛿 where 1- 𝛿 is rgfbCS. But, q1≀ rgfbCl(q1). This shows that, rgfbCl(q1) β‰  rgfbCl(q2). To prove the sufficiency: Let p1 & p2 be fuzzy singletons with various supports x1 & x2 respectively, q1 & q2 be crisp fuzzy singletons such that q1(x1)=1, q2(x2)=1. But, q1≀ rgfbCl(q1) οƒž 1-rgfbCl(q1) ≀ 1-q1≀1-p1. As each crisp fuzzy singleton is rgfbCS, 1- rgfbCl(q1) is rgfbOS and p2≀ 1- rgfbCl(q1) ≀ 1- p1.This proves, fts is rgfbT0 space. Definition 4.3: A fts is known as rgfbT1,that is regular generalized fuzzy bT1, iff for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs rgfbOSs 𝛿1 & 𝛿2 such that, q1≀ 𝛿1≀ 1- q2 and q2≀ 𝛿2≀ 1- q1. Theorem 4.4: A fts is rgfbT1, that is regular generalized fuzzy bT1, if and only if each crisp fuzzy singleton is rgfbCS. Proof: To prove the necessary condition: Let rgfbT1 be fts and crisp fuzzy singleton with supports x0 be q0 .There occurs, rgfbOSs 𝛿1 and 𝛿2 for any fuzzy singleton q with supports x (β‰  x0), such that, q0≀ 𝛿1≀ 1- q and q ≀ 𝛿2≀ 1- q0. Since, it includes each fuzzy set as the collection of fuzzy singletons. So that, 1-q0 = 01 qq V βˆ’ο‚£ q = 0. Thus, 1-q0 is rgfbOS. This shows that, q0 (crisp fuzzy singleton) is rgfbCS. Regular generalized fuzzy b-separation axioms in fuzzy topology 157 To prove the sufficiency: Assume p1 and p2 be pair of fuzzy singletons with various supports x1& x2 .Further on fuzzy singletons with various supports x1 & x2 be q1 & q2, such that q1(x1) = 1 and q2(x2)=1. As each crisp fuzzy singleton is rgfbCS, the fuzzy sets 1-q1 & 1-q2 are rgfbOSs such that, p1≀ 1- q1≀ 1- p2 and p2≀1-q2≀ 1- p1.This proves, fts is rgfbT1 space. Remark 4.5:In a fts X1, each rgfbT1 space is rgfbT0 space. Proof: It follows the above definition. The opposite of this theorem is in correct. This is shown as follows – Example 4.6:Let X1={a, b},p1={(a,0),(b,1)} and p2={(a,0.4),(b,0)} are fuzzy singletons. U= {(a, 0.5),(b, 1)} be rgfbOS . Let 𝜏= { 0,p1 , p2 , U,1 }. The space is rgfbT0 and it is not rgfbT1. Definition 4.7: A fts is known as rgfbT2 , that is regular generalized fuzzy bT2 or rgfb-Hausdorff iff, for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs, rgfbOS 𝛿1 & 𝛿2 such that, q1≀ 𝛿1≀ 1- q2 , q2≀ 𝛿2≀ 1- q1 and 𝛿1≀ 1- 𝛿2 . Theorem 4.8: A fts is known as rgfbT2, that is regular generalized fuzzy bT2 or rgfb-Hausdorff if and only if for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs an rgfbOS 𝛿1 such that, q1≀ 𝛿1≀ rgfbCl 𝛿1≀ 1- q2 . Proof: To prove the necessary condition: Let rgfbT1 be fts and fuzzy singletons q1 & q2 with various supports .Let 𝛿1 & 𝛿2 be rgfbOS such that, q1≀ 𝛿1≀ 1- q2 , q2≀ 𝛿2≀ 1- q1 and 𝛿1≀ 1- 𝛿2 where 1- 𝛿2 is rgfbCS. We have by definition, rgfbCl(𝛿1)=/\ {(1- 𝛿2) : (1- 𝛿2) rgfbCS} where 𝛿 1≀ 1- 𝛿2 .Also rgfbCl(𝛿1)β‰₯ 𝛿1.This shows that, q1≀ 𝛿1≀ rgfbCl (𝛿)1≀ 1- 𝛿2≀ 1- q2 οƒž q1≀ 𝛿1≀ rgfbCl (𝛿1)≀ 1- q2 . To prove the sufficiency: Assume q1 and q2 are pair of fuzzy singletons with various supports and 𝛿1 be rgfbOS. Since, q1≀ 𝛿1≀ rgfbCl (𝛿1)≀ 1- q2 οƒž q1≀ 𝛿1≀1- q2. Also q1≀rgfbCl( 𝛿1)≀ 1- q2 οƒž q2≀ 1- rgfbCl (𝛿1) ≀1-q1. This shows that, 1- rgfbCl (𝛿1) is rgfbOS. Also rgfbCl (𝛿1) ≀ 1- rgfbCl (𝛿2) . This proves that, fts is rgfbT2 space. Remark 4.9:In a fts X1,each rgfbT2 space is rgfbT1 space. Proof: It follows the above definition. The opposite of this theorem is in correct. This is shown as follows – Example 4.10: Let X1={a,b}. q1={(a, 0.2),(b, 0)} and q2={(a,0), (b,0.4)} are fuzzy singletons, O1= {(a,0.3),(b,0.4)} and O2= {(a,0.8),(b,0.7)} are rgfbOS .Let 𝜏 = { 0, p1 , p2 , O1 , O2, 1}. The space is rgfbT1 and it's not rgfbT2. Varsha Joshi and Dr.Jenifer J.Karnel 158 Definition 4.11: A fts is known as rgfbT2 1 2 , that is regular generalized fuzzy bT2 1 2 or rgfb-Urysohn iff for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs, rgfbOSs 𝛿1 & 𝛿2 such that, q1≀ 𝛿1≀ 1- q2 , q2≀ 𝛿2≀ 1- q1 and rgfbCl (𝛿1)≀ 1-rgfbCl (𝛿2) . Remark 4.12:In a fts X1,each rgfbT2 1 2 space is rgfbT2 space. Proof: It follows from the above definition. The opposite of this theorem is in correct. This is shown as follows – Example 4.13: Let X1={a, b}. q1={(a, 0.1),(b,0)} and q2={(a,0),(b,0.3)} are fuzzy singletons, O1= {(a,0.2),(b,0.3)} and O2= {(a,0.7),(b,0.5)} are rgfbOSs. Let 𝜏 ={ 0, p1 , p2 , O1 , O2, 1 }. The space is rgfbT2 and it's not rgfbT2 1 2 . Figure2. From the above definition and examples one can notice that the above chains of implication. Theorem 4.14: An injective function f: X1 β†’ X2 is rgfb-continuous, and X2 is fT0, then X1 is rgfbT0. Proof: Assume  & Ξ² be fuzzy singletons in X1 with various support then f () & f (Ξ²) belongs to X2, As f is injective and f ()β‰  f (Ξ²). As X2 is fT0, there occurs, a open set O in X2 such that, 𝑓() ≀ 𝑂 ≀ 1 βˆ’ 𝑓(𝛽)or 𝑓(𝛽) ≀ 𝑂 ≀ 1 βˆ’ 𝑓(), οƒž  ≀ 𝑓 βˆ’1(𝑂) ≀ 1 βˆ’ 𝛽 or 𝛽 ≀ 𝑓 βˆ’1(𝑂) ≀ 1 βˆ’ . Since, f : X1 β†’ X2 is rgfb-continuous, 𝑓 βˆ’1(𝑂) is rgfbOS in X1. This shows that, X1 is rgfbT0- space[ 4.1]. Theorem 4.15: An injective function f : X1 β†’ X2 is rgfb-irresolute, and X2 is rgfbT0, then X1 is rgfbT0. Proof: Assume  & Ξ² be fuzzy singletons in X1 with various support. As f is injective 𝑓() & 𝑓(𝛽) belongs to X2 and 𝑓() β‰  𝑓(𝛽). As, X2 is rgfbT0, there occurs rgfbOS O in X2 so that 𝑓() ≀ 𝑂 ≀ 1 βˆ’ 𝑓(𝛽) or 𝑓(𝛽) ≀ 𝑂 ≀ 1 βˆ’ 𝑓() οƒž  ≀ 𝑓 βˆ’1(𝑂) ≀ 1 βˆ’ 𝛽 or 𝛽 ≀ 𝑓 βˆ’1(𝑂) ≀ 1 βˆ’ . As, f is rgfb- irresolute 𝑓 βˆ’1(𝑂) is rgfbOS(X1). This shows that, X1 is rgfbT0 space[4.1 ]. Theorem 4.16:An injective function f : X1 β†’ X2 is strongly rgfb-continuous, and X2 is rgfbT0, then X1 is fT0. rgfbT2 𝟏 𝟐 rgfbT2 rgfbT1 rgfbT0 Regular generalized fuzzy b-separation axioms in fuzzy topology 159 Proof: Assume  & Ξ² be fuzzy singletons in X1 with various support. Since f is injective f () & f (Ξ²) belongs to X2 and 𝑓() β‰  𝑓(𝛽). As, X2 is rgfbT0, there occurs rgfbOS O in X2 so that, 𝑓() ≀ 𝑂 ≀ 1 βˆ’ 𝑓(𝛽) or 𝑓(𝛽) ≀ 𝑂 ≀ 1 βˆ’ 𝑓(), οƒž  ≀ 𝑓 βˆ’1(𝑂) ≀ 1 βˆ’ 𝛽 or 𝛽 ≀ 𝑓 βˆ’1(𝑂) ≀ 1 βˆ’ . Since, f is strongly rgfb-continuous, 𝑓 βˆ’1(𝑂) is fuzzy-open in X1. This shows that, X1 is fT0-space[ 2.8]. Theorem 4.17:An injective function f : X1 β†’ X2 is rgfb-continuous, and X2 is fT1, then X1 is rgfbT1. Proof: Assume  and Ξ² be fuzzy singletons in X1 with various supports. 𝑓() and 𝑓(𝛽) belongs to X2, Since, f is injective. As, X2 is fT1 space hence, by the statement there occurs fuzzy-open sets O1 & O2 in X2 such that, 𝑓() ≀ 𝑂1 ≀ 1 βˆ’ 𝑓(𝛽) and 𝑓(𝛽) ≀ 𝑂2 ≀ 1 βˆ’ 𝑓( ) οƒž  ≀ 𝑓 βˆ’1(𝑂1) ≀ 1 βˆ’ 𝛽 and 𝛽 ≀ 𝑓 βˆ’1(𝑂2) ≀ 1 βˆ’ . Since, f is rgfb-continuous 𝑓 βˆ’1(𝑂1) and 𝑓 βˆ’1(𝑂2) are rgfb-open in X1. This shows that, X1 is rgfbT1 space[4.3 ]. Theorem 4.18: An injective function f : X1 β†’ X2 is rgfb-irresolute, and X2 is rgfbT1, then X1 is rgfbT1. Proof: Assume  & Ξ² be fuzzy singletons in with various supports. Since f is injective, 𝑓() & 𝑓(𝛽) belongs to X2. As X2 is rgfbT1, there occurs two rgfbOS O1& O2 in X2 so that 𝑓() ≀ 𝑂1 ≀ 1 βˆ’ 𝑓(𝛽) and 𝑓(𝛽) ≀ 𝑂2 ≀ 1 βˆ’ 𝑓() οƒž  ≀ 𝑓 βˆ’1(𝑂1) ≀ 1 βˆ’ 𝛽 π‘Žπ‘›π‘‘ 𝛽 ≀ 𝑓 βˆ’1(𝑂2) ≀ 1 βˆ’ . Since, f is rgfb- irresolute, then 𝑓 βˆ’1(𝑂1) π‘Žπ‘›π‘‘ 𝑓 βˆ’1(𝑂2) are rgfbOS(X1). This shows that, X1 is rgfbT1 space[ 4.3]. Theorem 4.19:If f : X1 β†’ X2 is strongly rgfb-continuous and X2 is rgfbT1, then X1 is fT1. Proof: Assume  & Ξ² be fuzzy singletons in X1 with various supports. Since, f is injective, 𝑓() &𝑓(𝛽) belong to X2. As, X2 is rgfbT1, there occurs two rgfbOSs O1 and O2 in X2 so that, 𝑓() ≀ 𝑂1 ≀ 1 βˆ’ 𝑓(𝛽) and 𝑓(𝛽) ≀ 𝑂2 ≀ 1 βˆ’ 𝑓() οƒž  ≀ 𝑓 βˆ’1(𝑂1) ≀ 1 βˆ’ 𝛽 and 𝛽 ≀ 𝑓 βˆ’1(𝑂2) ≀ 1 βˆ’ . Since, f is strongly rgfb-continuous, therefore 𝑓 βˆ’1(𝑂1) & 𝑓 βˆ’1(𝑂2) are fuzzy-open in X1. This shows that, X1 is fT1 space[2.8 ]. Theorem 4.20: An injective function f : X1 β†’ X2 is rgfb-continuous, and X2 is fT2, then X1 is rgfbT2. Proof: Assume  & Ξ² be fuzzy singletons in X1 with various supports. Since, f is injective, so 𝑓() & 𝑓(𝛽) belongs to X2 and 𝑓() β‰  (𝛽). Since, X2 is fT2, therefore there occurs open fuzzy set O in X2 so that, 𝑓() ≀ 𝑂 ≀ Varsha Joshi and Dr.Jenifer J.Karnel 160 Cl(𝑂) ≀ 1 βˆ’ 𝑓(𝛽) οƒž  ≀ 𝑓 βˆ’1(𝑂) ≀ 𝑓 βˆ’1[𝐢𝑙(𝑂)] ≀ 1 βˆ’ 𝛽. Since, f is rgfb- continuous 𝑓 βˆ’1(𝑂) is rgfbCS(X1). Hence,  ≀ 𝑓 βˆ’1(𝑂) ≀ 𝑓 βˆ’1[Cl(𝑂)] ≀ 𝑓 βˆ’1[rgfbCl(𝑂)] ≀ rgfbCl[𝑓 βˆ’1[(𝑂)] ≀ 1 βˆ’ 𝛽. That is,  ≀ 𝑓 βˆ’1(𝑂) ≀ rgfbCl[𝑓 βˆ’1[(𝑂)] ≀ 1 βˆ’ 𝛽. This shows that, X1 is rgfbT2[4.7]. Theorem 4.21: An injective function f : X1 β†’ X2 is rgfb-irresolute, and X2 is rgfbT2. Then, X1 is rgfbT2. Proof: Obvious. Theorem 4.22: An injective function f : X1 β†’ X2 is strongly rgfb-continuous, and X2 is rgfbT2. Then, X1 is fT2. Proof: Obvious. Theorem 4.23: An injective function f : X1 β†’ X2 is rgfb-continuous, and X2 is fT2 1 2 . Then, X1 is rgfbT2 1 2 . Proof: Assume  & Ξ² be fuzzy singletons in X1 with various supports. Since, f is injective, then 𝑓() and 𝑓(𝛽) belongs to X2 and 𝑓 () β‰  𝑓(𝛽). Since, X2 is fT2 1 2 , then there occurs open fuzzy sets O1 and O2 in X2 such that, 𝑓() ≀ 𝑂1 ≀ 1 βˆ’ 𝑓(𝛽), 𝑓(𝛽) ≀ 𝑂2 ≀ 1 βˆ’ 𝑓()and Cl𝑂1 ≀ 1 βˆ’ Cl𝑂2 οƒž  ≀ 𝑓 βˆ’1(𝑂1) ≀ 1 βˆ’ 𝛽 ,𝛽 ≀ 𝑓 βˆ’1(𝑂2) ≀ 1 βˆ’  and Cl𝑓 βˆ’1(𝑂1) ≀ 1 βˆ’ Cl𝑓 βˆ’1(𝑂2). Since, 𝑓 is rgfb-continuous 𝑓 βˆ’1(𝑂1) and 𝑓 βˆ’1(𝑂2) are rgfbOS(X1). Cl(𝑓 βˆ’1(𝑂1)) ≀ rgfbCl(𝑓 βˆ’1(𝑂1)) and1 βˆ’ 𝐢l(𝑓 βˆ’1(𝑂2)) ≀ 1 βˆ’ rgfbCl(𝑓 βˆ’1(𝑂2)). Hence, rgfbCl(𝑓 βˆ’1(𝑂1)) ≀ 1 βˆ’ rgfbCl(𝑓 βˆ’1(𝑂2)). This shows that, X1 is rgfbT2 1 2 [ 4.11]. Acknowledgements The authors are grateful to principal of SDMCET, Dharwad and management SDM society for their support. Regular generalized fuzzy b-separation axioms in fuzzy topology 161 References [1] Azad,K. (1981). Fuzzy semi-continuity, Fuzzy Almost continuity and Fuzzy weakly continuity. Journal of Mathematics Analysis and Application,82, pp.14-32. [2] Balasubramaniam,G. and Sundaram. (1997). Some generalization of fuzzy continuous functions. Fuzzy Sets and Systems , 86(1), pp. 93-100. [3] Benchalli, S. and Karnel, J. (2010).On fuzzy b-open sets in fuzzy topological spaces. Journal of Computer and Mathematical Sciences, 1(2),pp.103-273. [4] Benchalli, S. and Karnel, J. (2010). 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