International Journal of Mathematical Archive-7(7), 2016, Available online through ISSN

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1 International Journal of Mathematical Archive-7(7), 2016, Available online through ISSN NANO SEMI GENEALIZED HOMEOMOPHISMS IN NANO TOPOLOGIAL SPAE K. BHUVANESWAI* 1, A. EZHILAASI 2 1Professor an Hea, Department of Mathematics, Mother Teresa Women s Universit Koaikanal, Tamilnau, Inia. 2esearch Scholar, Department of Mathematics, Karpagam Universit oimbatore, Tamilnau, Inia. (eceive On: ; eve & Accepte On: ) ABSTAT In th paper, we introuce nano sg-close maps an nano sg open maps in nano topological spaces an then we introuce an stuy nano sg - homeomorphms. We obtain certain characterizations of these maps. Keywors: Nano close map, Nano open map, Nano sg irresolute, Nano sg-close map, Nano sg-open map, nano sghomeomorphm. 1. INTODUTION The concepts of semi generalize mappings an generalize- semi mappings were introuce by.devi et.al [7] in 1993.Maki et.al [10] introuce g-homeomorphms an gc-homeomorphms in topological spaces. Devi [7] introuce an stuie sg- close functions an gs-close functions. A stuy on semi-generalize homeomorphm was one by Devi et. al [6]. The concept of nano topology was introuce by Lell Thivagar [7] an he analyse nano close maps, nano open maps an nano homeomorphms. In th paper, nano generalize semi close functions, nano semi-generalize open functions are introuce an nano semi-generalize homeomorphms are analyse. 2. PEMILINAIES Definition 2.1: [11] A function f : ( X, ) ( Y, σ ) calle semi-open if the image of every open set in ( X, ) semi-open in ( Y, σ ). Definition 2.2: [13] A function f : ( X, ) ( Y, σ ) calle g-open if the image of every open set in ( X, ) g-open in ( Y, σ ). Definition 2.3: [5] A function f : ( X, ) ( Y, σ ) calle pre-semi open if the image of every semi open set in ( X, ) semi-open in ( Y, σ ). Definition 2.4: [12] A function f : ( X, ) ( Y, σ ) calle sg-close if the image of every close set in ( X, ) sg close in ( Y, σ ). Definition: 2.5: [13] A function f : ( X, ) ( Y, σ ) semi-generalize continuous (sg-continuous) if sg-close set in X for every close set V of Y. f (V) Definition 2.6: [10] A bijective function f : ( X, ) ( Y, σ ) calle a generalize homeomorphm if f both g-continuous an g-open. orresponing Author: K. Bhuvaneswari* 1 1Professor an Hea, Department of Mathematics, Mother Teresa Women s Universit Koaikanal, Tamilnau, Inia. International Journal of Mathematical Archive- 7(7), July

2 K. Bhuvaneswari* 1, A. Ezhilarasi 2 / Nano Semi Generalize Homeomorphms in Nano Topological Space / IJMA- 7(7), July Definition 2.7: [6] A bijective function f : ( X, ) ( Y, σ ) calle a semi-generalize homeomorphm if f both sg-continuous an sg-open. Definition 2.8: [8] Let U be a non-empty finite set of objects calle the universe an be an equivalence relation on U name as incernibility relation. Then U ivie into joint equivalence classes. Elements belonging to the same equivalence class are sai to be incernible with one another. The pair (U, ) sai to be the approximation space. Let X U. Then, (i) The lower approximation of X with respect to the set of all objects which can be for certain classifie as X with respect to an enote by L (X). L (X) = U{(x): (x) X, xєu} where (x) enotes the equivalence class etermine by xєu. (ii) The upper approximation of X with respect to the set of all objects which can be possibly classifie as X with respect to an enote by U (X). U (X) = U{(x): (x) X Ф, xєu}. (iii) The bounary region of X with respect to the set of all objects which can be classifie neither as X nor as not X with respect to an it enote by B (X). B (X) = U (X) - L (X). Property 2.9 [8]: If (U, ) an approximation space an X, Y U, then 1. L (X) X U (X) 2. L (Ф) = U (Ф) = Ф 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) an U (X) U (Y) whenever X Y U (X c ) = [L (X)] c an L (X c ) = [U (X)] c 10. U [U (X)] = L [U (X)] = U (X) 11. L [L (X)] = U [L (X)] = L (X). Definition 2.10 [8]: Let U be the universe, be an equivalence relation on U an the nano topology (X) = {U, Ф, L (X), U (X), B (X)} where X U. Then by property 2.5, (X) satfies the following axioms: (i) U an Ф є (X). (ii) The union of the elements of any sub-collection of (X) in (X). (iii) The intersection of the elements of any finite subcollection of (X) in (X). Then (X) a topology on U calle the Nano topology on U with respect to X. (U, (X)) calle the Nano topological space. Elements of the Nano topology are known as nano open sets in U. Elements of [ (X)] c are calle nano close sets with [ (X)] c being calle Dual Nano topology of (X). If (X) the Nano topology on U with respect to X, then the set B = {U, L (X), B (X)} the bas for (X). Definition 2.11 [8]: If ( U, ( X )) a Nano topological space with respect to X where X U an if A U, then (i) The nano interior of the set A efine as the union of all nano open subsets containe in A an enote by NInt(. NInt( the largest nano open subset of A. (ii) The nano closure of the set A efine as the intersection of all nano close sets containing A an enote by Nl(. Nl( the smallest nano close set containing A. emark 2.12 [8]: Throughout th paper, U an V are non-empt finite universes; X U an Y V; U/ an V/ enote the families of equivalence classes by equivalence relations an respectively on U an V. ( U, ( X )) an (, ( Y )) are the Nano topological spaces with respect to X an Y respectively. V Definition 2.13 [1]: If ( U, ( X )) a Nano topological space with respect to X where X U an if A U, then (i) The nano semi-closure of A efine as the intersection of all nano semi-close sets containing A an enote by Nsl(. Nsl( the smallest nano semi-close set containing A an Nsl( A. (ii) The nano semi-interior of A efine as the union of all nano semi-open subsets of A an enote by NsInt(. NsInt( the largest nano semi open subset of A an NsInt( A. Definition 2.14 [1]: A subset A of ( U, ( X )) calle nano semi-generalize close set (Nsg-close) if NSl( V an A V an V nano semi-open in ( U, ( X )). 2016, IJMA. All ights eserve 169

3 K. Bhuvaneswari* 1, A. Ezhilarasi 2 / Nano Semi Generalize Homeomorphms in Nano Topological Space / IJMA- 7(7), July Definition 2.15 [1]: If ( U, ( X )) a nano topological space with respect to X where X U an if A U, then (i) The nano semi- generalize closure of A efine as the intersection of all nano semi-generalize close sets containing A an enote by Nsgl(. (ii) The nano semi-generalize interior of A efine as the union of all nano semi-generalize open subsets of A an enote by NsgInt(. Definition 2.16 [9]: Let ( U, ( X )) an ( V, ( Y )) be two nano topological spaces. Then a mapping ( X )) ( V, ( Y )) nano continuous on U if the inverse image of every nano open set in V nano open in U. Definition 2.17[9]: A function ( X )) ( V, ( Y )) calle nano open if the image of every nano open set in ( U, ( X )) nano open in (, ( Y )). V Definition 2.18 [9]: A function ( X )) ( V, ( Y )) calle nano close if the image of every nano close set in ( U, ( X )) nano close in (, ( Y )). V Definition 2.19 [4]: A function ( X )) ( V, ( Y )) calle nano g close if the image of every nano g close set in ( U, ( X )) nano g close in (, ( Y )). V Definition 2.20 [4]: A mapping ( X )) ( V, ( Y )) nano semi generalize continuous on U if the inverse image of every nano open set in V nano sg open in U. Definition 2.21 [9]: A bijection ( X )) ( V, ( Y )) calle nano homeomorphm if f both nano continuous an nano open. Definition 2.22 [4]: A bijection ( X )) ( V, ( Y )) calle nano g homeomorphm if f both nano g continuous an nano g open. 3. NANO SG-LOSED MAPS In th section, nano semi-generalize close functions an nano semi-generalize open functions in nano topological spaces are introuce. We cuss certain characterizations of these functions. Definition 3.1: Let ( U, ( X )) an ( V, ( Y )) be the two nano topological spaces. Then a function ( X )) ( V, ( Y )) sai to be nano semi generalize close function (briefly Nsg close function) if the image of every nano close set in ( U, ( X )) Nsg close in (, ( Y )). V Example 3.2: Let U = { a, } be the universe with U = { a},{ c},{ } an let X = { a, b} U. Then the nano open sets are ( X ) = { U, φ, { a},{ a, },{ } an the nano close sets are ( X ) = { U, φ, { a},{ a, },{ }.{ U,φ,{ a},{ a, c},{ },{ a, } are the nano semi open sets. Nsg open sets are { U,φ, { a},{ b},{ },{ a, b},{ a, c},{ a, },{ },{ a, c},{ a, },{ a, },{ }. Nsg close sets are { U,φ, { a},{ b},{ c},{ },{ c},{ },{ a, c},{ },{ a, },{ },{ a, c}}. Let V = { x, w} be the anotheruniverse with V = { x},{ w},{ z } an let Y = { x, z} V. Then the nano open sets are ( Y ) = { V, φ, { x},{ x, z},{ z } an the nano close sets are ( Y ) = { V, φ, { w},{ x, w},{ w } { V,φ,{ x},{ y},{ z},{ x, y},{ x, z},{ x, w},{ z},{ x, z},{ x, w},{ w},{ x, w } are the Nsg open sets. Nsg close sets are { V,φ,{ x},{ y},{ z},{ w},{ w},{ z},{ x, w},{ w},{ w},{ x, w},{ x, w }. Define the function ( X )) ( V, ( Y )) as f ( a) = f ( b) = x, f ( c) = w, f ( ) = z.now f ( U ) = V, f ( φ ) = φ, f ({ } ) = { x, w, z}, f ({ c} ) = { w}, f ({ a, c} ) = { w} are the images of nano close sets of U which are Nsg close in V. Thus the function ( X )) ( V, ( Y )) a Nsg close function. 2016, IJMA. All ights eserve 170

4 K. Bhuvaneswari* 1, A. Ezhilarasi 2 / Nano Semi Generalize Homeomorphms in Nano Topological Space / IJMA- 7(7), July Theorem 3.3: A function ( X )) ( V, ( Y )) Nsg close function if an only if Nsglf ( f ( Nl( for every subset A of ( U, ( X )). Proof: Let ( X )) ( V, ( Y )) be a Nsg close function an A U. Then Nl ( nano close in ( U, ( X )) an hence f ( Nl( Nsg close function in ( V, ( Y )). Since A Nl(, it implies that f ( f ( Nl(. As Nsgl ( f ( Nl( the Nsg close set containing f (, it follows that Nsgl ( f ( Nsgl ( f ( Nl( ) = f ( Nl(. onversel let A be any nano close set in ( U, ( X )). Then A = Nl( an so f ( = f ( Nl( Nsgl( f ( by the given hypothes. Also, it follows that f ( Nsgl( f (. Hence f ( = Nsgl( f (. i.e., f ( Nsg close an hence ( X )) ( V, ( Y )) a Nsg close function. Theorem 3.4: A function ( X )) ( V, ( Y )) Nsg close if an only if for each subset S of ( V, ( Y )) an for each nano open set A of ( U, ( X )) containing f 1 ( S ), there a Nsg open set B of (, ( Y )) such that S B an f ( B ) A. V Proof: Let S be the subset of ( V, ( Y )) an A be a nano open set of ( U, ( X )) such that f ( S ) A. Now V f ( U, say B, a Nsg open set containing S in V such that f ( B) A. onversel let F be a nano close set of ( U, ( X )), then f ( V f ( F)) U F an U F nano open. Now, there a Nsg open set B of ( V, ( Y )) such that V f ( F) B an f ( B) U F. Hence 1 F U f ( B) an thus V B f ( F) f ( U f ( B)) V B which implies f ( F) = V B. Since V B Nsg close, f (F) a Nsg close set in ( V, ( Y )) for every nano close set F in ( U, ( X )). Hence ( X )) ( V, ( Y )) a Nsg close function. The following example shows that the composition of two Nsg close functions nee not be Nsg close. Example 3.5: Let ( U, ( X )), ( V, ( Y )) an ( X )) ( V, ( Y )) be as in Example 3.2. Let W = { p, q, r, s} with W = { p},{ q},{ r, s }. Let Z = { p, r} W. The nano open sets are ( Z) = { W, ϕ, { p},{ prs,, },{ rs, }} an nano close sets are ( Z) = { W, φ, { q, r, s},{ q},{ p, q }. The Nsg close sets are { W,φ,{ p},{ r},{ q},{ s},{ p, q},{ r, q},{ r, s},{ q, s},{ p, r, q},{ p, q, s},{ q, r, s }. Define a map g : ( V, ( Y )) ( W, as g ( x) = r, y) = p, z) = q, w) = s. Then both f an g are Nsg close maps but their composition g ( X )) ( W, not a Nsg close map since for the nano close set { a, c} in ( U, ( X )), ( g f )({ a, c} ) = g[ f ({ a, c} )] = { w} ) = { p, s} which not a Nsg close set in (,. Thus the composition of two Nsg close maps nee not be Nsg close. W Theorem 3.6: Let ( X )) ( V, ( Y )) an g : ( V, ( Y )) ( W, be two mappings such that their composition g ( X )) ( W, a Nsg close mapping. Then the following statements are true. (i) If f nano continuous an surjective, then g Nsg close. (ii) If g Nsg irresolute an injective, then f Nsg close. (iii) If g strongly Nsg continuous an injective, then f nano close. 2016, IJMA. All ights eserve 171

5 K. Bhuvaneswari* 1, A. Ezhilarasi 2 / Nano Semi Generalize Homeomorphms in Nano Topological Space / IJMA- 7(7), July Proof: (i) Let A be a nano close set in ( V, ( Y )). Since ( X )) ( V, ( Y )) nano continuous, it follows that f ( A ) nano close in ( U, ( X )). Since g ( X )) ( W, Nsg close, ( g f )[ f 1 ( ] Nsg close in ( W,. i.e., g ( Nsg close in ( W, as the function ( X )) ( V, ( Y )) surjective. Hence the image of a nano close set in ( V, ( Y )) Nsg close in ( W,.Thus g : ( V, ( Y )) ( W, Nsg close. (ii) Let B be a nano close set in ( U, ( X )). Since the function g ( X )) ( W, Nsg close, then ( g f )( B) Nsg close in ( W,. Since g Nsg irresolute, g 1 [( g f )( B)] Nsg close in ( V, ( Y )) i.e., the set f (B) Nsg close in ( V, ( Y )) since the function g : ( V, ( Y )) ( W, injective. Thus the image of a nano close set in ( U, ( X )) Nsg close in (, ( Y )). Thus ( X )) ( V, ( Y )) Nsg close. V (iii) Let D be a nano close set in ( U, ( X )). Since the function g ( X )) ( W, Nsg close in ( W,, ( g f )( D) Nsg close in ( W,. Since the function g: ( V, ( Y)) ( W, strongly Nsg continuous, g 1 [( g f )( D)] nano close in ( V, ( Y )). i.e., since the functions g : ( V, ( Y )) ( W, injective, f (D) nano close in ( V, ( Y )) for every nano close set D in ( U, ( X )). Thus ( X )) ( V, ( Y )) nano close. Definition 3.7: A function ( X )) ( V, ( Y )) Nsg open in (, ( Y )) for each nano open set A in ( U, ( X )). V sai to be Nsg open function if the image f (A ) Definition 3.8: A function ( X )) ( V, ( Y )) Nsg open function if an only if f ( NInt( NsgInt( f ( for each subset A of ( U, ( X )). Proof: Suppose that ( X )) ( V, ( Y )) be a Nsg open function. Let A U. Then NInt ( nano open in ( U, ( X )). Since ( X )) ( V, ( Y )) Nsg open, f ( NInt( Nsg open in ( V, ( Y )). Also, as NInt( A, it follows that f ( NInt( f (. Thus f ( NInt( Nsg open containe in f (. Hence f ( NInt( NsgInt( f (. onversel let f ( NInt( NsgInt( f ( for every subset A of ( U, ( X )). Let F be a nano open set in ( U, ( X )), then NInt ( F) = F. Now f ( F) f ( NInt( F)) NsgInt( f ( F)) = f ( F). Hence f ( F) = NsgInt( f ( F)). Thus f (F) Nsg open in ( V, ( Y )) for every nano open set F in ( U, ( X )). Hence ( X )) ( V, ( Y )) a Nsg open function. 4. NANO SG-HOMEOMOPHISMS In th section, a new form of homeomorphms namel Nsg homeomorphm introuce an some of the properties are stuie. Definition 4.1 A function ( X )) ( V, ( Y )) sai to be a Nsg homeomorphm if (i) f one to one an onto (ii) f a Nsg continuous (iii) f a Nsg open Theorem 4.2: Let ( X )) ( V, ( Y )) be an one to one onto mapping. Then f Nsg homeomorphm if an only if f Nsg close an Nsg continuous. 2016, IJMA. All ights eserve 172

6 K. Bhuvaneswari* 1, A. Ezhilarasi 2 / Nano Semi Generalize Homeomorphms in Nano Topological Space / IJMA- 7(7), July Proof: Let ( X )) ( V, ( Y )) be a Nsg homeomorphm. Then f Nsg continuous. Let A be an arbitrary nano close set in ( U, ( X )). Then U A nano open. Since f Nsg open, f ( U Nsg open in ( V, ( Y )). That, V f ( Nsg open in ( V, ( Y )). Hence f ( Nsg close in ( V, ( Y )) for every nano close set A in ( U, ( X )). Hence ( X )) ( V, ( Y )) Nsg close. onversel let f be Nsg close an Nsg continuous function. Let G be a nano open set in ( U, ( X )). Then U G nano close in ( U, ( X )). Since f Nsg close, f ( U G) Nsg close in ( V, ( Y )). That, f ( U G) = V f ( G) Nsg close in ( V, ( Y )). Hence f (G) Nsg open in ( V, ( Y )) for every nano open set G in ( U, ( X )). Thus f Nsg open an hence ( X )) ( V, ( Y )) Nsg homeomorphm. Theorem 4.3: A one to one map f of ( U, ( X )) onto ( V, ( Y )) a Nsg homeomorphm if an only if f ( Nsgl( = Nl( f ( for every subset A of ( U, ( X )). Proof: If ( X )) ( V, ( Y )) Nsg homeomorphm, then f Nsg continuous an Nsg close. If A U, it follows that f ( Nsgl( Nl( f ( since f Nsg continuous. Since Nsgl( nano close in ( U, ( X )) an f Nsg close map, f ( Nsgl( Nsg close in (, ( Y )). Also 2016, IJMA. All ights eserve 173 V Nsgl ( f ( Nsgl( ) = f ( Nsgl( f ( ). Since A Nsgl(, f ( f ( Nsgl( an hence Nl ( f ( Nl( f ( Nsgl( ) = f ( Nsgl(. Thus Nl( f ( f ( Nsgl(. Therefore, f ( Nsgl( = Nl( f ( if f Nsg homeomorphm. f ( Nsgl( = Nl( f ( A for every subset A of ( U, ( X )), then f Nsg continuous. If A nano close in ( U, ( X )), then A Nsg close in ( U, ( X )). Then Nsgl ( = A which implies f ( Nsgl( = f (. Hence, by the given hypothes, it follows that Nl ( f ( = f (. Thus f ( onversel if )) nano close in V an hence Nsg close in V for every nano close set A in U. That, f Nsg close. Thus ( X )) ( V, ( Y )) Nsg homeomorphm. Example 4.4: Let U = { a, } with U = { a},{ c},{ } an let X { a b} U ( X ) = { U, φ, { a},{ a, },{ }. Let V = { x, w} with V = { x},{ w},{ z } an let Y { x z} V The nano open sets are ( Y ) = { V, φ, { x},{ x, z},{ y z } ( X )) ( V, ( Y, =,. The nano open sets are =,.. Define a function )) as f ( a) = f ( b) = x, f ( c) = w, f ( ) = z Then f bijective, Nsg continuous an Nsg open an so the function ( X )) ( V, ( Y )) Nsg homeomorphm. Theorem 4.5: If a function ( X )) ( V, ( Y )) nano homeomorphm, then it Nsg homeomorphm but not conversely. Proof: As the function ( X )) ( V, ( Y )) nano homeomorphm, by efinition, f bijective, nano continuous an nano open. An hence ( X )) ( V, ( Y )) Nsg continuous. Since ( X )) ( V, ( Y )) nano open, the image of every nano open set in ( U, ( X )) nano open in ( V, ( Y )) an hence Nsg open in ( V, ( Y )) every nano open set Nsg open. Thus the function ( X )) ( V, ( Y )) Nsg open. Therefore, every nano homeomorphm ( X )) ( V, ( Y )) Nsg homeomorphm. The converse of the Theorem 4.5 nee not be true as seen from the following example. Example 4.6: In Example 4.4, the map f a Nsg homeomorphm. Now f ( V ) = U, ( φ ) = φ f ( x) = b, f 1 ({ z} ) = { a, }, f 1 ({ x, z} ) = { a, }. Hence the inverse images of nano open sets in ( V, ( Y )) are not nano open in ( U, ( X )) an hence f not nano continuous. Thus the map ( X )) ( V, ( Y )) not nano homeomorphm. f, { }

7 K. Bhuvaneswari* 1, A. Ezhilarasi 2 / Nano Semi Generalize Homeomorphms in Nano Topological Space / IJMA- 7(7), July Theorem 4.7: Let ( X )) ( V, ( Y )) be a Nsg continuous function. Then the following statement are equivalent. (i) f a Nsg open function (ii) f a Nsg homeomorphm (iii) f a Nsg close function. Proof: (i) (ii): By the given hypothes, the function ( X )) ( V, ( Y )) bijective, Nsg continuous an Nsg open. Hence the function ( X )) ( V, ( Y )) Nsg homeomorphm. (ii) (iii): By the given hypothes, the function ( X )) ( V, ( Y )) Nsg homeomorphm an hence Nsg open. Let A be the nano close set in ( U, ( X )). Then A nano open in ( U, ( X )) By assumption, f ( A ) Nsg open in ( V, ( Y )).i.e., f ( A ) = ( f ( Nsg open in ( V, ( Y )) an hence f ( Nsg close in ( V, ( Y )) for every nano close set A in ( U, ( X )). Hence the function ( X )) ( V, ( Y )) Nsg close function. (iii) (i): Let F be a nano open set in ( U, ( X )). Then F nano close set in ( U, ( X )). By the given hypothes, f ( F ) Nsg close in ( V, ( Y )). Now, f ( F ) = ( f ( F)) Nsg close, i.e., f (F) Nsg open in ( V, ( Y )) for every nano open set F in ( U, ( X )). Hence ( X )) ( V, ( Y )) Nsg -open function. The composition of two Nsg -homeomorphms nee not always be a Nsg -homeomorphm as seen from the following example. Example 4.8: Let ( U, ( X )), (, ( Y )) an (, be three nano topological spaces an let V W = V = W = { a, c }, then the nano open sets are ( X ) { U, φ, { a},{ a, },{ } ( Y ) { V, φ, { b},{ a, c},{ a c } an ( Z) = { W, φ, { c},{ a, c},{ a b } U, =, =,,. Define two functions ( X )) ( V, ( Y )) an g : ( V, ( Y )) ( W, as f ( a) = f ( b) = f ( c) = a, f ( ) = an g ( a) = a, b) = c) = ) =. Here the functions f an g are Nsg continuous an bijective. Also the image of every nano open set in ( U, ( X )) Nsg open in ( V, ( Y )). i.e., f ({ a, } ) = { a, c}, f ({ } ) = { a, c}, f ({ a} ) = { b}. Thus the function ( X )) ( V, ( Y )) Nsg open an thus Nsg homeomorphm. The map g : ( V, ( Y )) ( W, also Nsg continuous, bijective an Nsg open. Hence g also Nsg homeomorphm. But their composition g ( X )) ( W, not a Nsg homeomorphm because for the nano open set F = { a, b} in ( W,, ( g f ) 1 ( F) = f ( g ({ a, b} )) = f 1 ({ a, b} ) = { a, } not Nsg open in ( U, ( X )). Hence the composition g ( X )) ( W, not Nsg continuous an thus not a Nsg -homeomorphm. Thus the composition of two Nsg homeomorphms nee not be a Nsg -homeomorphm. EFEENES 1. K.Bhuvaneswari an A.Ezhilarasi, On Nano semi-generalize an Nano generalize semi close sets in Nano Topological Spaces, International Journal of Mathematics an omputer Applications esearch, (2014), K.Bhuvaneswari an A.Ezhilarasi, On Nano Semi-Generalize ontinuous Maps in Nano Topological Spaces, International esearch Journal of Pure Algebra 5(9), 2015, K.Bhuvaneswari an K. Mythili Gnanapriya, Nano generalize close sets in nano topological spaces. International Journal of Scientific an esearch Publications, 4(5): K.Bhuvaneswari an K. Mythili Gnanapriya, On Nano generalize continuous function in nano topological space. International Journal of mathematical Archive, 6(6): , IJMA. All ights eserve 174

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