PROPERTIES OF NANO GENERALIZED- SEMI CLOSED SETS IN NANO TOPOLOGICAL SPACE

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1 Volume 116 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v116i12.18 ijpam.eu POPETIES OF NANO GENEALIZED- SEMI LOSED SETS IN NANO TOPOLOGIAL SPAE A.EZHILAASI 1 and K. BHUVANESWAI 2 1Assistant Professor, Department of Mathematics, Kumaraguru ollege of Technology, oimbatore, Tamilnadu, India. 2 Professor and Head, Department of Mathematics, Mother Teresa Women s University, Kodaikanal, Tamilnadu, India. Abstract The properties of a new class of sets, namely nano generalized-semi closed sets in nano topological space are analysed in this paper. The relation of these sets with already existing well known sets are studied. Keywords: Nano g-closed sets, Nano gs-open sets, Nano gs-closed sets, Nano gsclosure, Nano gs-interior, Nano gs- neighbourhood. 1. Introduction The concept of generalized-semi closed sets to characterize the S-normality axiom was introduced by S.P.Arya et.al (1). The semi-generalized mappings and generalized-semi mappings were studied by.devi et.al (6) in detail. In 2013, Govindappa Navalagi investigated some of the regularity axioms, normality axioms and continuous functions through gs-open sets and sg-open sets. Also, Govindappa Navalagi (8) continued the study of gs-continuous and sg-continuous functions to introduce the new notions like generalized semiclosure and generalized semi-interior operators. Lellis Thivagar (9) obtained the notion of nano topology and he studied the various forms of nano sets, their closures and interiors and their homeomorphisms. 2. Nano Generalized-Semi losed Sets 167

2 In this section, a new form of Ng -closed sets called nano generalized-semi closed sets in ( U, ( )) is introduced and an analysis of certain basic properties of these sets is done. Definition: 3.1 For a subset A of ( U, ( )), if Nsl( A) V, A V and V is nano open in ( U, ( )), then A is called nano generalizedsemi closed set (briefly Ngs -closed). The subset A is called Ngs -open in ( U, ( )) if A is Ngs - closed. Further the family of all Ngs -open sets in ( U, ( )) Ngs is denoted by ( ). Definition: 3.2 The intersection of all Ngs -closed sets containing A is defined as the nano generalized-semi closure of A, denoted by Ngsl(A) which is the smallest Ngs -closed set containing A. Also, Ngsl( A) A.and thus Ngsl( A) A if A is Ngs -closed set. Definition: 3.3 The union of all Ngs -open subsets of A is named as the nano generalized-semi interior of A, denoted by NgsInt (A). NgsInt (A) is the largest Ngs -open subset of A. Also, NgsInt ( A) A. If A is Ngs -open, then NgsInt( A) A. Example:3.4 Let U c, d be the universe with U a c, d let, b. Then ( ) U, d d, and which are nano open sets. The nano closed sets are given by d c c U,,. The nano semi-closed sets are given by d, c, d c U,,. The nano semi-open sets are U,, c, d b, d d,. Hence, Ngs -closed sets are given by U,, b, c, d, a, b, c, a, c, d c, d, a, c c d d, and thus the Ngs -open sets are U,, d, b c, d, d c d d d,. 168

3 emark:3.5 If A is a Nsg -closed set, then it is Ngs -closed. The following example shows that the converse of the emark 3.5 need not always be true. Example:3.6 Let U c be the universe with U a, c.then ( ) U,, and let which are nano open sets. The nano closed sets are ( ) U, c sets are U,, c c, c b.the nano semi-closed,,.the nano semi-open sets are U,,. Thus, Ngs -closed sets are given by c, c, c b U,, and Ngs -open sets are U, c, c, b. Nsg -closed sets are U, c, c. Then Nsg -open sets are given by U,, c, b every Ngs -closed set need not be a Nsg -closed.,. Hence, 3. Nano Generalized-Semi losure And The Basic Properties In this section, nano generalized-semi closure of a set is defined and some of the properties are analysed. Definition :4.1 The Ngs -closure of A is defined as the intersection of all nano generalized-semi closed sets containing A, denoted by Ngsl (A) which is the smallest Ngs - closed set containing A. If A is Ngs -closed, then Ngsl( A) A. Further Ngsl( A) A. Definition :4.2 The Ngs - interior of A is defined as the union of all nano generalized-semi open subsets of A and is denoted by NgsInt( A) which is the largest Ngs and if A is Ngs -open, then NgsInt( A) A. In general, NgsInt( A) A. emark :4.3 For any set E U, E Ngsl ( Nsl( Nl(. In Example 3.4, let E d U. Now Ngsl d ( and 169

4 ( c d and Nl c, d Nsl, E Ngsl ( Nsl( Nl( (.Thus it follows that emark :4.4 Both containment relations in the emark 4.3 may be proper as seen from the following example. Example :4.5 In Example 3.4, let E, b U. Now Ngsl c ( c d. Also Nsl c, d Nl, ( and ( and hence it follows from the results that E Ngsl ( Nsl( Nl(. emark :4.6 For any subset A of a nano topological space ( U, ( )), the following properties are true. (i) Ngsl ( U A) U NgsInt ( A) (ii) NgsInt ( U A) U Ngsl ( A) Example :4.7 With U, U and as in Example 3.4, let A U be such that b and thus U A c, d A,. (i) Now Ngsl ( U A) Ngsl ( c, d) c, d, b) b and NgsInt ( A) NgsInt (,. Further, b c d U NgsInt( A) U,. Hence Ngsl ( U A) U NgsInt ( A). (ii) Now NgsInt U A) NgsInt ( c, d) d, b), b c ( and Ngsl ( A) Ngsl (,. It follows that, b c d U Ngsl( A) U,. Hence NgsInt ( U A) U Ngsl ( A). Definition :4.8 A subset A of a nano topological space ( U, ( )) is called Ngs -neighbourhood (briefly Ngs -nbd) of a point x of U if there exists Ngs -open set F containing x such that x F A. Theorem :

5 A subset of a nano topological space ( U, ( )) is Ngs -open if and only if it is a Ngs -nbd of each of its points. Proof. Let F be a subset of ( U, ( )). Let F be a Ngs -open set and let x F. Now F is a Ngs -open set containing a point x. Hence F is a Ngs -nbd of each of its points. onversely, let A be a Ngs -nbd of each of its points. Hence A is Ngs -open set containing each of its points and A is Ngs -open. Definition :4.10. A point x of U is called a nano generalized-semi interior (briefly Ngs -interior) point of A U if and only if x A. Theorem :4.11 Let A be a subset of a nano topological space ( U, ( )) and let x U. Then x is a Ngs -interior point of A if and only if A is a Ngs -nbd of x. Proof. Let A U and x U. Let x be the Ngs -interior point of A. Hence x A and by the given hypothesis, there exists a Ngs - open set F containing x such that Ngs -nbd of x. x F A.Hence A is the onversely, let A be a Ngs -nbd of x. Given A U, x U and hence there exists a Ngs -open set F containing x such that x A F. Since x F, a Ngs -open set such that A F, x A, it follows that x is the Ngs -interior point of A. Theorem :4.12 Let A be a subset of ( U, ( )). Then x Ngsl( A) if and only if for any Ngs -nbd Proof. N of x in ( U, ( )), A N. Assume x Ngsl( A). Let N be a Ngs -nbd of the point x in ( U, ( )) such that A N. Since N is a Ngs -nbd of x in ( U, ( )), there exists a Ngs -open set V such that 171

6 x V N. Hence, it follows that V A and so A ( ) V. Since ( V ) is a Ngs -closed set containing A, and hence x Ngsl( A) which is a contradiction. onversely, let Ngsl ( A) ( ) V N of x be a Ngs -nbd of the nano topological space ( U, ( )) such that A N. Let us prove by the method of contradiction. Suppose that x Ngsl( A), then there exists a Ngs -closed set F of ( U, ( )) such that A F and x F.Thus x F and F is Ngs -open in ( U, ( )) and hence F is a Ngs -nbd of x in ( U, ( )) which is a contradiction. Thus x Nsgl( A). Theorem :4.13 A subset A of ( U, ( )). But A F is Ngs -closed if and only if Nsl( A) A contains no non empty nano closed set. Proof. Let a Ngs -closed set be taken as A and let F be a nano closed subset of Nsl( A) A. Then it follows that A F. Then the nano open set F is nano semi-open. Since A is Ngs -closed, then it implies that Nsl ( A) F. onsequently F ( Nsl( A)) and also F Nsl( A). Hence by the implication that F Nsl( A) ( Nsl( A)), it follows that the nano closed subset F is empty. onversely, let Nsl( A) A contains no non-empty nano closed set. Let A F and F be a nano open set. Suppose that Nsl( A) F (by contradiction), then Nsl ( A) F. Since Nsl (A) is a nano closed set and F is nano closed in ( U, ( )), Therefore Nsl ( A) F is also nano closed set in ( U, ( )). Nsl( A) F Nsl( A) A and so Nsl( A) A contains a non-empty nano closed set which is a contradiction to the given hypothesis. Hence it follows that Nsl( A) F. Thus A is Ngs -closed. 172

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