Characterisation of Nano generalized β closed sets in Nano topological spaces

Transcription

1 IJSA, 4(1), 2017; International Journal of Sciences & Applied esearch wwwijsarin haracterisation of Nano generalized β sets in Nano topological spaces S B Shalini, K Indirani* Department of Mathematics, Nirmala ollege for Women, oimbatore, Tamil Nadu, India orrespondence Address: *K Indirani, Department of Mathematics, Nirmala ollege for Women, oimbatore, Tamil Nadu, India Abstract The objective of this paper is to introduce a new class of sets called Nano generalized β sets in Nano topological spaces and also investigate the characteristic of these defined sets Keywords: Nano β interior, Nano β closure and Nano generalized β sets 1 Introduction Levine [7] introduced generalized sets as a generalization of sets in topological spaces Andrijevic [1] introduced a new class of generalized open sets in a topological space, so called semi pre open sets, later Dontchev [4] studied generalized semi pre open sets and the equivalent notion of Nano open sets was discussed by Gnanambal [5] The notion of Nano topology is introduced by Lellis Thivagar [6] which is defined in terms of approximations and boundary regions of a subset of an universe using an equivalence relation on it and he also defined Nano set, Nano interior and Nano closure The aim of this paper is to continue the study of nano generalized sets in nano topological space In particular, we introduce a new class of nano sets on Nano topological spaces called Nano generalized sets and obtain their characteristics counter examples 7 2 Preliminaries Definition: 21[6]Let U be a non-empty finite set of objects called the universe and be an equivalence relation on U named as indiscernibility relation Then U is divided into disjoint equivalence classes Elements belonging to the same equivalence class are said to be indiscernible one another The pair U, is said to be the approximation space Let X U Then, The lower approximation of X respect to is the set of all objects which can be for certainly classified as X respect to and is denoted L X by L x: x x X, x U where denotes the equivalence class determined by x U The upper approximation of X respect to is the set of all objects which can be possibly classified as X respect to and is denoted by U

2 x x U : X, x U The boundary region of X respect to is the set of all objects which can be classified neither as X nor as not X respect to and is denoted by B B U L Property: 22 [6] If U, is an approximation space and X, Y U, then 1) L X U 2) L U 3) L U U U U 4) U Y U U Y 5) U Y U U Y 6) L Y L L Y 7) L Y L L Y 8) L L Y and U U Y whenever X Y 9) L L U U and 10) U U L U U 11) L L U L L Definition: 23[6] Let U be the universe, be an equivalence relation on U and U,, L, U, B wher e X U Then by property 22 satisfies the following axioms: i) U and ii) The union of the elements of any sub collection of is in iii) The intersection of the elements of any finite sub collection of is in Then is a topology on U called the Nano topology on U respect to X, U, is called the Nano topological space Elements of the Nano topology are known as Nano open sets in U Elements X are called Nano sets of 8 IJSA, 4(1), 2017; X being called dual Nano topology of emark: 24[6] If is the Nano topology on U respect to X, then the set B U, L, B is the basis X for Definition: 25[6] If U, is a Nano topological space respect X where X U and if A U, then i)the Nano interior of a set A is defined as the union of all Nano open subsets contained in A and is denoted by N int A N intaisthe largest Nano open subset of A ii)the Nano closure of a set A is defined as the intersection of all Nano sets containing A and is denoted by Ncl A Ncl Ais the smallest Nano set containing A Definition: 26 [2][5][7]Let U, be a Nano topological space and A U Then A is said to be (i) Nano semi open if A Ncl N int A A (ii) Nano pre open if A N int Ncl (iii) Nano open if A N intncl N int A (iv) Nano regular open if A N int Ncl A (v) Nano b open if A Ncl N int A N int Ncl A (vi) Nano open(nano semi-pre open) if A Ncl N int Ncl A NSO U, X, NPO U, X, N OU, X, NO ( U, X ), NBO( U, X ) and N OU, X respectivelyenote the families of all Nano semi open, Nano pre open, Nano open, Nano regular open,

3 Nano b open and Nano open subsets ofu Let U, be a Nano topological space and A U Then A is said to benano semi, Nano pre, Nano, Nano regular, Nano b and Nano if its complement is respectively Nano semi open, Nano pre open, Nano open, Nano regular open, Nano b open and Nano open Definition: 27 [3] A subset A of U, is called Nano generalized set (briefly Ng ) if Ncl A V whenever A V and V is U, X Nano open in 3 Nanogeneralized sets Definition: 31If U, is a Nano topological space respect X where X U and if A U, then i)the Nano interior of a set A is defined as the union of all Nano open subsets contained in A and is denoted by N inta N intais the largest Nano open subset of A ii)the Nano closure of a set A is defined as the intersection of all Nano sets containing A and is denoted by N cla NclAis the smallest Nano set containing A Definition: 32A subset A of Nano topological space U, is called Nano generalized set (briefly Ng ) if N cla V whenever A V and V is Nano open in U, Theorem: 33Every Nano set in U, is Nano g set U, X in IJSA, 4(1), 2017; Proof: Assume that A is a Nano set in U, and let V is Nano open in U, such that A V, N cla A V That is N cla V Therefore A is Nano g set emark: 34The converse of the above theorem need not be true which can be seen from the following example Example: 35Let U a d U a, b, c, d and X a as U,, a Here the seta, dis Nano g but not Nano inu The following theorem can also be proved in a similar way Theorem: 36Let U, be a Nano topological space and A U Then (i) Every Nano set is Nano g set (ii) Every Nano semi set is Nano g set (iii) Every Nano pre set is Nano g set (iv) Every Nano set is Nano g set (v) Every Nano regular set is Nano g set (vi) Every Nano b set is Nano g set (vii) Every Nano g set is Nano g set (viii) Every Nano gs set is Nano g set (ix) Every Nano g set is Nano g set (x) Every Nano g r set is Nano g set 9

4 emark: 37everse implications of the above theorem 36 need not be true which can be seen from the following example Example: 38Let U a d, e U a, b, d, e and X a, b as U,, a, a d, b, d Here the set a cis Nano g but not Nano, Nano semi, Nano pre, Nano, Nano regular, Nano b, Nano g, Nano g s, Nano g, Nano g r inu 4 haracterisation of nano generalized sets Theorem: 41The union of two Nano g sets need not be Nano g set which can be seen from the following example Example: 42Let U a d U a, c, b and X a, b Then the Nano topology is defined X U,, a, a, b, d, b d Here as, IJSA, 4(1), 2017; the sets a andbare Nano g sets buta b a d is not Nano g set inu Theorem: 43The intersection of two Nano g sets need not be Nano g set which can be seen from the following example Example: 44Let U a d U a, b, c, d and X a as U,, a Here the sets a, b d and a, dare Nano g a d a, d a, d is not sets but Nano g set inu Theorem: 45Let A be a in U, then N cla A Ng set does not contain any non-empty Nano set U, X in We have the following implications for the properties of subsets: Nano semi Nano Nano pre Nano b Nano gs Nano Nano g Nano Nano g Nano gr 10

5 Proof:Let A be a Ng set in U, and F be Nano subsets of N cla A That is F N cla A F N cl A A That is implies F NclAand F A which implies A F where F is a Nano open set Since A is Nano g, N cla F That is F Ncl A Thus F NclA NclA, F Hence N cla A does not contain any U, X non-empty Nano set in Theorem: 46Let A be a Ng set in U, then A is Nano if and only if N cla A is Nano set in U, Proof:Let A be a Ng set Assume that A is Nano then we have NclA A, N cla A Hence NclA Ais Nano onversely, assume that NclA Abe Nano Then by theorem 45 NclA Adoes not contain any nonempty set Thus N cla A That is N cla A Therefore is Nano Theorem: 47If A is Nano g in U, and B is any set in U, such that A B NclA, then B is also Nano g in U, Proof: Let A be a Ng set ofu such that A B NclA Let B V where V be Nano open set in U then A V Since A is Ng, we have N cla V B Ncl A, Now 11 IJSA, 4(1), 2017; N clb NclNclA NclA V, N clb V Therefore B is Nano g set inu Theorem: 48Let A be Nano open and Nano g set in U, then A F is Nano g whenever F NclU, X Proof: Let A be Nano open and Nano g set then NclA A and A NclA Therefore N cla A Thus A is Nano Hence A F is Nano in U which implies that A F is Nano g inu eferences 1 Andrijevic D, Semi-pre open sets, Mat Vesnik, 38(1) (1986), Arockiarani I, and Arokia Lancy, A, Generalized soft gβ sets and soft gsβ sets insoft topological spaces, International Journal of Mathematical Archive, 4(2) (2013), Bhuvaneswari K and Mythili Gnanapriya K, Nano generalised sets in nano topological spaces, International Journal of Scientific and esearchpublications,4(5)(2014)1-3 4 Dontchev J, On generalizing semipreopen sets, Mem Fac Sci Kochi Univ Ser A Math 16(1995), Gnanambal Y, On Nano open sets, Int Jr of Engineering, 1(2) (2015), Lellis Thivagar M, and armel ichard, On Nano forms of weakly open sets, International Journal of Mathematics and Statistics Invention, 1(1) (2013), Levine N, Generalized sets in topology, end irc Mat Palermo 19(2) (1970), 89-96