ON NANO REGULAR GENERALIZED STAR b-closed SET IN NANO TOPOLOGICAL SPACES

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1 An Open Aess, Online International Journal Available at eview Artile ON NANO EGULA GENEALIZED STA b-closed SET IN NANO TOPOLOGICAL SPACES *Smitha M.G. and Indirani K. Department of Mathematis, Nirmala College for Women, Coimbatore, Tamil Nadu, India *Author for Correspondene ABSTACT The purpose of this paper is to introdue a new lass of nano generalized losed sets namely, Nano regular generalized star b-losed sets in Nano topologial spaes and some of its basi properties are analyzed. Keywords: Nano rg * -losed set, Nano rg * -open set and nano rg * b-losed set INTODUCTION In 1970, Levine introdued the onept of generalized losed sets as a generalization of losed sets in topologial spaes. Ahmed and Mohd (2009) studied the lass of generalized b-losed sets. Sindhu and Indirani (2013) introdued the onept of regular generalized star b-losed sets in topologial spaes. The notion of Nano topology was introdued by Lellis whih was defined in terms of approximations and boundary regions of a subset of an universe using an equivalene relation on it and also defined Nano losed set, Nano interior and Nano losure. The aim of this paper is to ontinue the study of nano generalized losed sets in nano topologial spae. In partiular, we introdue a new lass of nano sets on Nano topologial spaes alled Nano regular generalized star b-losed sets and obtain their harateristis with ounter examples. Preliminaries Definition 2.1 [8]: Let U be a non-empty finite set of objets alled the universe and be an equivalene relation on U named as indisernibility relation. Then U is divided into disjoint equivalene lasses. Elements belonging to the same equivalene lass are said to be indisernible with one another. The pair (U,) is said to be the approximation spae. Let X U. Then, The lower approximation of X with respet to is the set of all objets whih an be for ertain lassified as X with respet to and is denoted by L (X). L (X) = {(x): (x) X, x U} where (x) denotes the equivalene lass determined by x U. The upper approximation of X with respet to is the set of all objets whih an be possibly lassified as X with respet to and is denoted by U (X). U (X) = {(x): (x) X φ, x U} The boundary region of X with respet to is the set of all objets whih an be lassified neither as X nor as not X with respet to and is denoted by B (X). B (X) = U (X) L (X) Property 2.2 [8]: If (U, ) is an approximation spae and X, Y U, then 1) L (X) X U (X) 2) L (φ) = U (φ) = φ 3) L (U) = U (U) = U 4) U (X Y) = U (X) U (Y) 5) U (X Y) U (X) U (Y) 6) L (X Y) L (X) L (Y) 7) L (X Y) = L (X) L (Y) 8) L (X) L (Y) and U (X) U (Y) whenever X Y Copyright 2014 Centre for Info Bio Tehnology (CIBTeh) 22

2 An Open Aess, Online International Journal Available at eview Artile 9) U (X ) = [L (X)] and L (X ) = [U (X)] 10) U (U (X)) = L (U (X)) = U (X) 11) L (L (X)) = U (L (X)) = L (X) Definition 2.3 [8]: Let U be the universe, be an equivalene relation on U and τ (X) = {U, φ, L (X), U (X), B (X)} where X U. Then by property 2.2 τ (X) satisfies the following axioms: i) U and φ τ (X). ii) The union of the elements of any sub olletion of τ (X) is in τ (X). iii) The intersetion of all elements of any finite sub olletion of τ (X) is in τ (X). Then τ (X) is a topology on U alled the Nano topology on U with respet to X, (U, τ (X)) is alled the Nano topologial spae. Elements of the Nano topology are known as Nano open sets in U. Elements of [τ (X)] are alled Nano losed sets with [τ (X)] being alled dual Nano topology of τ (X). emark 2.4 [8]: If τ (X) is the Nano topology on U with respet to X, then the set B = {U, L (X), B (X)} is the basis for τ (X). Definition 2.5 [8]: If (U, τ (X)) is a Nano topologial spae with respet X where X U and if A U, then The Nano interior of a set A is defined as the union of all Nano open subsets ontained in A and is denoted by NInt(. NInt( is the largest Nano open subset of A. The Nano losure of a set A is defined as the intersetion of all Nano losed sets ontaining A and is denoted by NCl(. NCl( is the smallest Nano losed set ontaining A. Definition 2.6 [8]: Let (U, τ (X)) be a Nano topologial spae and A U. Then A is said to be Nano semi open if A NCl(NInt( Nano pre open if A NInt(NCl( Nano α open if A NInt[NCl(NInt(] Nano regular open if A = NInt(NCl( NSO(U, X), NPO(U, X), τ α (X) and NO(U, X) respetively denote the families of all Nano semi open, Nano pre open, Nano α open, and Nano regular open subsets of U. Let (U, τ (X)) be a Nano topologial spae and A U, A is said to be nano semi-losed, nano prelosed, nano α-losed and nano regular losed if its omplement is respetively Nano semi open, Nano pre open, Nano α open, and Nano regular open. Definition 2.7 [3], [4]: A subset A of (U, τ (X)) is alled (i) Nano b-losed set (briefly Nb-losed) if NCl( NInt( NInt( NCl( A. (ii) Nano generalized losed set (briefly Ng-losed) if NCl( V whenever A V and V is Nano open in (U, τ (X)). (iii) Nano generalized b-losed set (briefly Ngb-losed) if NbCl( V whenever A V and V is Nano open in (U, τ (X)). (iv) Nano regular generalized b-losed set (briefly Nrgb-losed) if NbCl( V whenever A V and V is Nano regular open in (U, τ (X)). Nano egular Generalized Star b-losed Set Definition 3.1: Let ( U, be a nano topologial spae and A U. Then A is said to be regular generalized star losed set (briefly Nrg * -losed) if open in ( U,. Nl ( U whenever A U U is nano regular Let (U, τ (X)) be a Nano topologial spae and A U, A is said to be nano regular generalized star open set (briefly Nrg * -open) if its omplement is Nrg * -losed. Copyright 2014 Centre for Info Bio Tehnology (CIBTeh) 23

3 An Open Aess, Online International Journal Available at eview Artile Definition 3.2: Let ( U, be a nano topologial spae and A U. Then A is said to be regular Nbl ( U whenever A U U is Nrg * - generalized star b-losed set (briefly Nrg * b-losed) if open in ( U,. Nrg * O(U,X) denotes the family of all nano rg * -open subsets of U. Example 3.3: Let U= {a,b,,d} with U / ={{a},{b,},{d}} and X = {a,b}. Then the nano topology (X ) {{a}, {a,b,}, {b,}, U, }. The nano losed sets are U,, {b,,d}, {d} and {a,d}.then Nrg * O(U,X) = {{a}, {b}, {}, {d}, {a,b}, {b,}, {a,}, {b,d}, {,d},{a,b,}, {a,b,d}, U, }. Then rg * b- losed sets are {a}, {b}, {}, {d}, {b,}, {a,d}, {b,d}, {,d}, {b,,d}, {a,b,d}, {a,,d}, U and. Theorem 3.4: Every nano losed set is nano rg * b-losed. Proof: Let A be a nano losed set in ( U, and G be nano rg * -open suh that A G. Sine every nano losed set is nano b-losed, Nbl ( Nl ( A Therefore A is nano rg * b- losed. emark 3.5: Converse of the above theorem need not be true as seen from the following example. Example 3.6: Let U = {a,b,,d} with U / = {{a}, {b,}, {d}} and X = {a,b}. Then the nano topology (X ) {{a}, {a,b,}, {b,}, U, }. Then the sets {b,} and {b,d} are rg * b-losed, but not losed. The following theorems an also be proved in a similar way. Theorem 3.7: Let ( U, be a nano topologial spae and A U. Then (i) Every nano semi-losed set is nano rg * b-losed. (ii) Every nano pre-losed set is nano rg * b-losed. (iii) Every nano losed set is nano rg * b-losed. (iv) Every nano regular losed set is nano rg * b-losed. emark 3.8: everse impliations of the theorem need not be true as seen from the following example. Example 3.9: Let U= {x,y,z,w} with U / ={{x,y}, {z}, {w}} and X = {x,z}. Then the nano topology (X ) {{z}, {x,y}, {x,y,z}, U, }. Here the sets {x} and {y} are nano rg * b-losed, but not nano semi-losed. The set {z} is nano rg * b-losed, but not nano pre-losed. Example 3.10: Let U = {a,b,,d} with U / = {{a}, {b,}, {d}} and X = {a,b}. Then the nano topology (X ) {{a}, {a,b,}, {b,}, U, }. The sets {b} and {b,} are rg * b-losed, but not nano Also the sets {b,} and {a,d} are rg * b-losed, but not nano regular losed. Theorem 3.11: Let ( U, (i) Every nano rg * b-losed set is nano rgb-losed. (ii) Every nano rg * b-losed set is nano gb-losed. Proof: Let A be a nano rg * b-losed set in ( U, be a nano topologial spae and A U. Then Copyright 2014 Centre for Info Bio Tehnology (CIBTeh) 24 losed. and G be nano regular open suh that A G. Sine every nano regular open set is nano rg * -open, G is nano rg * -open. Sine A is nano rg * b-losed, Nbl ( Therefore A is nano rgb-losed. (ii) The proof is similar. emark 3.12: The onverse of the theorem need not be true as seen from the following example. Example 3.13: Let U= {a,b,,d,e} with U / ={{a,e}, {b,d}, {}} and X = {,d}. Then the nano topology (X ) {{}, {b,d}, {b,,d}, U, }. Here the set {b,}is nano rgb-losed, but not nano rg * b- losed and the set {a,b,}is nano gb-losed, but not nano rg * b-losed emark 3.14: The nano rg * b-losed sets are independent of the nano losed sets like nano g-losed sets and nano rg * -losed sets as shown by the following example.

4 An Open Aess, Online International Journal Available at eview Artile Example 3.15: Let U= {a,b,,d,e} with U / ={{a,e}, {b,d}, {}} and X = {,d}. Then the nano topology (X ) {{}, {b,d}, {b,,d}, U, }. Here the set {b,}is nano rg * -losed, but not nano rg * b- losed. The set {b}is nano rg * b -losed, but not nano rg * -losed. The set {a,b,}is nano g-losed, but not nano rg * b-losed. The set {b,d}is nano rg * b -losed, but not nano g-losed. Theorem 3.16: The union of two nano rg * b-losed sets need not be nano rg * b-losed as seen from the following example. Example 3.17: Let U= {a,b,,d,e} with U / ={{a,e}, {b,d}, {}} and X = {,d}. Then the nano topology (X ) {{}, {b,d}, {b,,d}, U, }. Here the sets {b} and {} are nano rg * b-losed, but their union { b} { } { b, } not nano rg * b-losed. Theorem 3.18: Let A be a nano rg * b-losed set in ( U, ( X)). Then Nbl( - A has no non-empty nano rg * -losed set. Proof: Let A be a nano rg * b-losed. Let F be a nano rg * -losed set in Nbl( - A. That is, F Nbl( A F Nbl( A. F A A F, where F is a nano rg * -open set. Sine A is nano rg * b-losed, Nbl( F. That is, F ( Nbl(. Thus F Nbl( ( Nbl(. Therefore F. Theorem 3.19: Let A be nano rg * b-losed set in ( U, ( X)). Then A is nano b-losed iff Nbl( - A is nano rg * -losed. Proof: Let A be a nano rg * b-losed set. If A is nano b-losed, we have Nbl( - A =, whih is nano rg * -losed. Conversely let Nbl( - A is nano rg * -losed. Then by theorem 3.18, Nbl( - A =, whih implies Nbl( = A. Therefore A is nano b-losed. Theorem 3.20: If Ais nano rg * b-losed set and B is any set suh that A B Nbl(, then B is nano rg * b-losed. Proof: Let B G, where G is a nano rg * -open set. Sine A is nano rg * b-losed and A G, Nbl ( Also Nbl( B) Nbl( Nbl( Nbl( Thus Nbl ( B) G and hene B is nano rg * b-losed. Theorem 3.21: If A is nano rg * -open and nano rg * b-losed, then A is nano b-losed. Proof: Let A be nano rg * -open and nano rg * b-losed. Then ( A, A Nbl(. Therefore A = Nbl(. Hene A is nano b-losed. Copyright 2014 Centre for Info Bio Tehnology (CIBTeh) 25 Nbl but always EFEENCES Andrijevi D (1996). On b-open sets. Matematički Vesnik Ahmad Al-Omari and Mohd. Salmi Md. Noorani (2009). On Generalized b-losed sets. Bulletin of the Malaysian Mathematial Sienes Soiety (2) 32(1) Dhanis Arul Mary A and Arokiarani I (2015). On Nano gb-closed sets in Nano Topologial spaes. International Journal of Mathematial Arhieve Dhanis Arul Mary A and Arokiarani I (2015). On Charaterizations of Nano rgb-closed sets in Nano Topologial spaes. International Journal of Modern Engineering esearh Indirani K and Sindhu G (2013). On egular Generalized star b-losed set. International Journal of Mathematial Arhieve 4(10) Levine N (1070). Generalized lose sets in topology. endionti del Cirolo Matematio di Palermo 19(2)

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