Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962 Volume-7, Issue-1, January-February 2017 International Journal of Engineering and Management Research Page Number: 49-51 Nano Star Pre Generalized Continuous Functions in Nano Topological Spaces Nancy J E1, Vaiyomathi K2 PG Student, Nirmala College for Women, Coimbatore, Tamil Nadu, INDIA 2 Assistant Professor, Nirmala College for Women, Coimbatore, Tamil Nadu, INDIA 1 ABSTRACT The purpose of this paper is to introduce, Nano star pre generalized continuous function in Nano Topological Spaces. Keywords-- Nano topological space, Nano*pg closed, Nano*pg continuous function I. INTRODUCTION Continuous function is one of the main concepts of Topology. Balachandran et al. [2] and Mashouret al. [3] have introduced g-continuous and precontinuous function in topological spaces respectively. Arokiarani [1] introduced generalized pre-continuous functions and generalized pre-irresolute functions and compared with various stronger forms of the same functions. The notion of Nano topology was introduced (i) by LellisThivagar[5] which was defined in terms of (ii) approximations and boundary region of a subset of an (iii) universe using an equivalence relation on it and he also (iv) defined Nano closed sets, Nano-interior, Nano-closure,(v) Nano continuous functions, Nano open mapping, Nano(vi) closed mapping and Nano Homeomorphism. Nancy et(vii) al[4] introduced and studied some properties of Nano star pre generalized closed sets in Nano topological spaces. In this paper, a new class of continuous functions(ix) called Nano star pre generalized continuous function is(x) introduced and some of its properties in terms of Ng – closed sets, Ng-closure and Ng- interior are discussed. II. PRELIMINARIES (i) Definition:2.1:[6]A subset A of (U, ) is called(ii) Nano pre generalized closed set (briefly Npg-closed) if Npcl(A)⊆V whenever A⊆V and V is Nanopre open in(iii) (U, ). Definition:2.2[7] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Then U is divided into disjoint equivalence classes. Elements 49 belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space. Let X⊆ U 1. The lower approximation of X with respect to R is the set of all objects, which can be for certainly classified as X with respect to R and it is denoted by . That is ⊆ } , where R(x) denotes ⋃ { the equivalence class determined by X U. 2. The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by . That is } ⋃ { 3. The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R and it is denoted by . That is Definition:2.3[7] If (U,R) is an approximation space and X,Y⊆U, then ⊆ ⊆ ⊆ ⊆ (viii) [ [ ] . ⊆ ] ⊆ Definition:2.4[7] Let U be non-empty, finite universe of objects and R be an equivalence relation on U. Let X⊆U. { }. If Let satisfies the following axioms: . is in . The intersection of the elements of any finite sub collection of is in Then is a topology on U called the nano topology on with respect to X. The elements of are called as nano-open sets. Copyright © 2016. Vandana Publications. All Rights Reserved. www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962 Definition:2.5[7]:A subset of (U, ) is called Nanogeneralized closed set (briefly Ng-closed) if Ncl(A)⊆V whenever A⊆V and V is Nano open set in (U, ). Definition:2.6[4] A subset A of (U, ) is called Nano*pre generalized closed set (briefly N*pg closed) if Nint(Npcl(A))⊆V whenever A⊆V and V is Nano pre open in (U, ). Definition:2.7[3] Let (U, and (V, be two nano topological spaces. Then a mapping f: (U, (V, is nano continuous function on U if the inverse image of every nano open set in (V, is nano open in (U, . Definition:2.8[9] Let (U, and (V, be two nano topological spaces. Then a mapping f: (U, (V, is nano generalized continuous function (shortly Ng-continuous) on U if the inverse image of every nano open set in (V, is nano generalized open in (U, . III. PROPERTIES OF NANO STAR PRE GENERALIZED CONTINUOUS FUNCTION IN NANO TOPOLOGICAL SPACE Definition:3.1 Let (U, and (V, be two nano topological spaces. Then a mappingf:(U, (V, is nano*pre generalized continuous (shortly N*pg continuous) function on U if the inverse image of every nano open set in (V, is nano*pre generalized open in (U, . Example:3.2 Let U={a,b,c,d} with U/R={{a},{c},{b,d}} and X={a,b}. Then ={U, {a,b,d},{b,d}}. Then nano* pre generalized open sets are: {U, ,{a},{b},{c},{d},{a,b},{b,c},{c,d}{a,d}, {b,c},{a,c},{a,b,c},{a,b,d},{b,c,d}{a,c,d}}.Let V={x,y,z,w} with V/R´={{x},{y,z},{w}} and Y={x,z}. Then {V, { } { },{y,z}} and ={ V, {y,z,w},{w},{x,w}} Define: f: (U, (V, as f(a)=x, f(b)=y, f(c)=w, { } { } f(d)=z Then { } That is the inverse image of every nano open set in V is nano* pre generalized open sets in U. Therefore f is nano* pre generalized continuous. Theorem:3.3 Every nano continuous function is nano*pre generalized continuous. Proof: Let f: (U, (V, be nano continuous on (U, . Since f in nano continuous of (U, ), the inverse image of everynano open set in (V, is nano open in (U, . But every nano open set is nano*pre generalized open set. Hence the inverse image of every nano open set (V, is nano*pre generalized open in (U, . Therefore, f is nano*pre generalized continuous. Remark:3.4 The converse of the above theorem is not true as seen from the following example. 50 Example:3.5 Let U={a,b,c}=V. Then ={U, {a,c}} with U/R={{a,c},{b}} and X={a,c} { } { }} with V/R´={{b},{a,c}} and { and Y={a,b} Define f: (U, (V, as f(a)=c, f(b)=b, which is nano*pre generalized continuous. But for the nano open set {a,c} in (V, its inverse { } is not nano open in (U, .Hence f is not nano continuous. Theorem:3.6 Every nano pre continuous function is nano*pre generalized continuous. Proof: Let f: (U, (V, be nano pre continuous on (U, . Since f in nano pre continuous of (U, ), the inverse image of every nano open set in (V, is nano open in (U, . But every nano pre open set is nano*pre generalized open set. Hence the inverse image of every nano open set (V, is nano*pre generalized open in (U, . Therefore, f is nano*pre generalized continuous. Remark:3.7 The converse of the above theorem is not true as seen from the following example. Example:3.8 Let U={a,b,c}=V. Then ={U, {a}} with U/R={{a},{b,c}} and X={a} and { } { }}with V/R´={{b},{a,c}} and Y={a,b} { Define f: (U, (V, is an identity mapping. Then f is nano*pre generalized continuous. But for the nano closed set {b} in (V, its inverse { } is not nano pre open in (U, . Hence f is not nano pre continuous. Theorem:3.9 Every nano generalized continuous function is nano*pre generalized continuous. Proof: Let f: (U, (V, be nano generalized continuous on (U, . Since f in nano generalized continuous of (U, ), the inverse image of every nano open set in (V, is nano open in (U, .But every nano generalized open set is nano*pre generalized open set.Hence the inverse image of every nano generalized open set (V, is nano*pre generalized open in (U, .Therefore, f is nano*pre generalized continuous. Remark:3.10 The converse of the above theorem is not true as seen from the following example. Example:3.11Let U={a,b,c}=V. Then ={U, {a,c}} with U/R={{a,c},{b}} and X={a,c} { } { }} with V/R´={{b},{a,c}} and { and Y={a,b} Define f: (U, (V, as f(a)=c, f(b)=b, which is nano*pre generalized continuous. But for the nano open set {a,c} in (V, its inverse { } is not nano generalized open in (U, .Hence f is not nano generalized continuous. Theorem:3.12 A function f: (U, (V, is nano*pre generalized continuous iff the inverse image of every nano open set in (V, is nano*pre generalized open in (U, . Proof: Let f: (U, (V, be nano*pre generalized continuous and F be nano open in (V, . Then is nano closed in (V, . Since f is nano*pre generalized continuous. Copyright © 2016. Vandana Publications. All Rights Reserved. www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962 is nano*pre generalized closed in .But Therefore, is nano*pre generalized open in (U, .Thus the inverse image of every nano open set in (V, is nano*pre generalized open in (U, if f is nano*pre generalized continuous on (U, . Conversely, Assume that is nano*pre generalized open in (U, for each nano open set in (V, Let G be a nano closed set in (V, . Then is nano open in (V, and by assumption, is nano*pre generalized continuous on (U, . Since = we have is nano*pre generalized continuous on (U, . Therefore f nano*pre generalized continuous. (U, REFERENCES [1] Arokirani, I., K.Balachandran& Julian Dontchev, 1999. Some characterizations of gp-irresolute and gpcontinuous maps between topological spaces.Mem.Fac.Sci.Kochi Univ. (Math.), 20: 93-104. [2] Balachandran, K., P. Sundaram & H.Maki, 1991.On generalized continuous maps in topological spaces. Mem. Fac. Sci. Kochi Univ. Ser. A, Math., 12: 5-13. [3] Bhuvaneswari,K. and K.MythiliGnanapriya, 2014. Nano Generalized closed sets in Nano Topological spaces. International Journal of Scientific and Research Publications., 4(5): 1-4 51 [4] Nancy.J.E and Vaiyomathi.K, 2016. Nano star pre generalized closed sets in Nano topological spaces. Mathematical Sciences International Research Journal, Vol.5.Issue.2: 2278-8697. [5]. LellisThivagar, M. and Carmel Richard, 2013. On Nano forms of weakly open sets. International Journal of Mathematics and Statistics Invention, 1(1): 31 -37. [6] H.Maki ,J.Umehara and T.Noiri , Every Topological space is pre -�1/2, Mem.Fac.Sci.Kochi Univ. Ser.A . math.,17(1996), 33-42 [7] M.LellisThivagar, and Carmel Richard, On Nano forms of weakly open sets,, International Journal of Mathematics and Statistics Invention, Volume Issue 1,August 2013,Pp.31 -37. [8] LellisThivagar, M. and Carmel Richard, 2013.On Nano Continuity. Mathematical Theory and Modeling., 3(7): 32 [9] Bhuvaneswari,K. and K.MythiliGnanapriya, 2015. On Nano Generalized continuous function in Nano Topological Spaces. International Journal of Mathematical Archive., 6 (6) : 182-186 Copyright © 2016. Vandana Publications. All Rights Reserved.